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THE  IMPROVED  SLATED  ARITHMETIC. 

Entered  according  to  Act  Of  Congress,  in  the  year  1872,  by  A.  S.  BARNES  &  Co.,  in  the  Office  of  the 
Librarian  of  Congress,  at  Washington. 

SILICATE    BOOK    SLATE    SURFACE.      Patented  February  24,  1S57  ;  January  15,  1867;  ami 

August  25,  1868. 

JOCELYN'S   SLATED   BOOK.      Patent  applied  for. 
BARNES'  SLATE  AND  WATERPROOF   FLY-LEAF   COMBINATION.      Patent  applied  fur. 


SCHOOL 


V 


ARITHMETIC. 


ANALYTICAL  AND  PRACTICAL. 


BY  CHARLES  DAVIES,  LL.D., 

[99*  DAVIES'  PRACTICAL  ARITHMETIC,  OF  THE  NEW  SERIES,  WITH  FULL  MODERN  TRXAT> 
KENT  OF  THE  SUBJECT,  IS  OF  THE  SAME  GRADE,  AND  DESIGNED  TO  TAKE  THK  PLACE  OF 
THIS  WORK.] 


A.  S.   BARNES    &   COMPANY, 

NEW  YORK,  CHICAGO  AND  NEW  ORLEANS, 


A  NEW  SERIES  OF  MATHEMATICS, 

By   CHARLES    DAVIES,    LL.D., 

AUTHOR     OF     THE     WEST     POINT     COURSE     OF     MATHEMATICS, 


The  following  named  volumes  are  entirely  new  works,  written  within  the  past 
ten  years,  to  conform  to  all  modern  improvement,  and  take  the  place  of  the 
author's  older  series. 

NO    CONFLICT    OP  EDITIONS 

is  possible,  if  patrons  will  be  particular  to  order  the  book  they  want  by  its  exact 
title.  Whenever  any  change  is  made  so  radical  as  to  be  likely  to  cause  confusion 
in  classes, 

THE   NAME    OF    THE    BOOK  IS    CHANGED. 

Teachers  using  any  work  by  DAVIES  not  here-in-after  enumerated,  are  not 
availing  themselves  of  the  advantages  offered  by 

THE    NEW    SERIES. 

{3^°  Primary,  Intellectual,  and  Practical  A  rithmetics  constitute  the  Series 
proper.  Other  volumes  are  optional. 

DAVIES'    PRIMARY    ARITHMETIC. 

The  elementary  combinations,  by  object  lessons. 

DAVIES'  INTELLECTUAL  ARITHMETIC. 

Referring  all  processes  to  the  Unit  for  analysis. 

DAVIES'  ELEMENTS  OF  WRITTEN  ARITH. 

Prominently  practical,  with  few  rules  and  explanations. 

DAVIES'     PRACTICAL     ARITHMETIC. 

Complete  theory  and  practice.     Substitute  for  this  volume. 

DAVIES'     UNIVERSITY    ARITHMETIC. 

A  purely  scientific  presentation  for  advanced  classes. 


DAVIES'  NE\gKNT|jr  ALGEBRA. 

A  connectmgiiBhbetweeBPrithmetic  and  Algebra. 

AND    A    FULL 

COURSE  OF  HIGHER  MATHEMATICS. 


Entered  according  to  Act  of  Congress,  in  the  year  1852,  by 
CHARLES     DAVIES, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern 
District  of  New  York. 

N.  S.  A. 


PREFACE. 


ARITHMETIC  embraces  the  science  of  numbers,  together  with  all  th« 
rules  which  are  employed  in  applying  the  principles  of  this  science 
to  practical  purposes.  It  is  -the  foundation  of  the  exact  and  mixed 
sciences,  and  the  first  subject,  in  a  well-arranged  course  of  instruc- 
tion, to  which  the  reasoning  powers  of  the  mind  are  directed.  Because 
of  its  great  practical  uses  and  applications,  it  has  become  the  guide 
and  daily  companion  of  the  mechanic  and  man  of  business.  Hence, 
a  full  and  accurate  knowledge  of  Arithmetic  is  one  of  the  most  im- 
portant elements  of  a  liberal  or  practical  education. 

Soon  after  the  publication,  in  1848,  of  the  last  edition  of  my  School 
Arithmetic,  it  occurred  to  me  that  the  interests  of  education  might  be 
promoted  by  preparing  a  full  analysis  of  the  science  of  mathematics, 
and  explaining  in  connection  the  most  improved  methods  of  teaching. 
The  results  of  that  undertaking  were  given  to  the  public  under  the 
title  of  "Logic  and  Utility  of  Mathematics,  with  the  best  methods  of  in- 
struction explained  and  illustrated."  The  reception  of  that  work  by 
teachers,  and  by  the  public  generally,  is*,  strong  proof  of  the  deep  interest 
which  is  felt  in  any  effort,  however  humble,  which  may  be  made  to 
improve  our  systems  of  public  instruction. 

In  that  work  a  few  general  principles  are  laid  down  to  which  it  is. 
supposed  all  the  operations  in  numbers  may  be  referred  : 

1st.  The  unit  1  is  regarded  as  the  base  bfjjfary  number,  and  the 
consideration  of  it  as  the  first  step  in  the  analysis  of  every  question 
relating  to  numbers. 

2d.  Every  number  is  treated  as  a  collection  of  units,  or  as  made  up 
of  sets  of  such  collections,  each  collection  having  its  own  base,  which 
is  either  1,  or  some  number  derived  from  1. 

'3d.  The  numbers  expressing  the  relation  between  the  different  units 
of  a  number  are  called  the  SCALE;  and  the  employment  of  this  term 
enables  us  to  generalize  the  laws  which  regulate  the  formation  of 
numbers. 

4th.  By  employing  the  term  "fractional  units"  the  same  principles 
are  made  applicable  to  fractional  numbers ;  for,  all  fractions  are  but 
collections  of  fractional  units,  these  units  having  a  known  relation  to  I. 


M306011 


IV  PREFACE. 

In  the  preparation  of  this  work,  two  objects  have  been  kept  con- 
etantly  in  view: 

1st.  To  make  it  educational ;  and, 
2d.  To  make  it  practical. 
To  attain  these  ends,  the  following  plan  has  been  adopted : 

1.  To  introduce  every  new  idea  to  the  mind  of  the  pupil  by  a  sim- 
ple question,  and  then  to  express  that  idea  in  general  terms  under  the 
form  of  a  definition. 

2.  When  a  sufficient  number  of  ideas  are  thus  fixed  in  the  mind, 
they  are  combined  to  form  the  basis  of  an  analysis;  so  that  all  the 
principles  are  developed  by  analysis  in  their  proper  order. 

3.  An  entire  system  of  Mental  Arithmetic  has  been  carried  forward 
with  the  text,  by  means  of  a  series  of  connected  questions  placed  at 
the  bottom  of  each  page;  and  if  these,  or  their  equivalents,  are  care- 
fully put  by  the  teacher,  the  pupil  will  understand  the  reasoning  in 
every  process,  and  at  the  same  time  cultivate  the  powers  of  analysis 
and  abstraction. 

4.  The  work  has  been  divided  into  sections,  each  containing  a  num- 
ber of  connected  principles ;  and  these  sections  constitute  a  series  of 
dependent  propositions  that  make  up  the  entire  system  of  principles 
and  rules  which  the  work  develops. 

Great  pains  have  been  taken  to  make  the  work  PRACTICAL  in  its 
general  character,  by  explaining^ind  illustrating  the  various  applica- 
tions of  Arithmetic  in  the  transactions  of  business,  and  by  connecting 
as  closely  as  possible,  every  principle  or  rule,  with  all  the  applications 
which  belong  to  it. 

I  have  great  pleasure  in  acknowledging  my  obligations  to  many 
teachers  who  have  favored  me  with  valuable  suggestions  in  regard  to 
the  definitions,  rules,  and  methods  of  illustration,  in  the  previous  edi- 
tions. I  hope  they  will  find  the  present  work  free  from  the  defects 
they  have  so  kindly  pointed  out 

A  Key  to  this  volume  has  been  prepared  for  the  use  of  Teachers  onty 


CONTENTS. 


JTRST  FIVE  RULES. 

Definitions. , 9—10 

Notation  and  Numeration . . . .' 10 — 22 

Addition  of  Simple  Numbers 22—30 

Applications  in  Addition 30 — 33 

Subtraction  of  Simple  Numbers 33—37 

Applications  in  Subtraction 37 — 42 

Multiplication  of  Simple  Numbers 42 — 50 

Factors 50—53 

Applications : 53 — 56 

Division  of  Simple  Numbers 56 — 61 

Equal  parts  of  Numbers 61 — 64 

Long  Division 64 — 68 

Proof  of  Multiplication 68—69 

Contractions  in  Multiplication 69—71 

Contractions  in  Division 71 — 74 

Applications  in  the  preceeding  Rules 74 — 79 

UNITED  STATES  MONET. 

United  States  Money  defined w 79 

Table  of  United  States  Money 79 

Numeration  of  United  States  Money 80 

Reduction  of  United  States  Money 81—83 

Addition  of  United  States  Money 83—85 

Subtraction  of  United  States  Money 85 — 87 

Multiplication  of  United  States  Money 87—91 

Division  of  United  States  Money 91 — 93 

Applications  in  the  Four  Rules 93 — 96 

DENOMINATE  NUMBERS. 

English  Money 96—  97 

Reduction  of  Denominate  Numbers 97 —  99 

Linear  Measure 99 — 101 

Cloth  Measure 101 — 102 

Land  or  Square  Measure 102—104 


VI  CONTENTS. 

Cubic  Measure  or  Measure  of  Volume 104 — 106 

Wine  or  Liquid  Measure *.'.  106—108 

Ale  or  Beer  Measure 108—109 

Dry  Measure 109—110 

Avoirdupois  Weight 110—111 

Troy  Weight 111—112 

Apothecaries'  Weight 112—114 

Measure  of  Time 114—116 

Circular  Measure  or  Motion 116 

Miscellaneous  Table 117 

Miscellaneous  Examples 117 — 1 19 

Addition  of  Denominate  Numbers 1 19 — 124 

Subtraction  of  Denominate  Numbers 124 — 125 

Time  between  Dates 125 

Applications  in  Addition  and  Subtraction 126 — 128 

Multiplication  .of  Denominate  Numbers 128 — 130 

Division  of  Denominate  Numbers 130—134 

Longitude  and  Time 134 

PROPERTIES  OF  NUMBERS. 

Composite  and  Prime  Numbers 135 — 137 

Divisibility  of  Numbers 137 

Greatest  Common  Divisor 137—140 

Greatest  Common  Dividend 140—142 

Cancellation 142—145 

COMMON  FRACTIONS. 

Definition  of,  and  First  Principles 146—149 

Of  the  different  kinds  of  Common  Fractions 149—150 

Six  Fundamental  Propositions <• 150 — 154 

Reduction  of  Common  Fractions 154 — 161 

Addition  of  Common  Fractions 161—162 

Subtraction  of  Common  Fractions 162 — 164 

Multiplication  of  Common  Fractions  164—168 

Division  of  Common  Fractions : 168—172 

Reduction  of  Complex  Fractions 172 

Denominate  Fractions 173—176 

Addition  and  Subtraction  of  Denominate  Fractions 176 — 178 

DUODECIMALS. 

Definitions  of,  &c 178—180 

Multiplication  of  Duodecimals 180—182 


CONTENTS.  VII 

DECIMAL  FRACTIONS. 

Definition  of  Decimal  Fractions r 182 — 183 

Decimal  Numeration — First  Principles 183 — 187 

Addition  of  Decimal  Fractions 187 — 191 

Subtraction  of  Decimal  Fractions 191—193 

Multiplication  of  Decimal  Fractions 193 — 195 

Division  of  Decimal  Fractions 195—197 

Applications  in  the  Four  Rules 197 — 198 

Denominate  Decimals 198 

Reduction  of  Denominate  Decimals 198—201 

ANALYSIS. 

General  Principles  and  Methods 201—213 

RATIO  AND  PROPORTION. 

Ratio  defined 213—214 

Proportion ,  214—216 

Simple  and  Compound  Ratio 216—218 

Single  Rule  of  Three 218—223 

Double  Rule  of  Three 223—228 

APPLICATIONS  TO  BUSINESS. 

Partnership 228—229 

Compound  Partnership 229—231 

Percentage 231—234 

Stock  Commission  and  Brokerage 234—237 

Profit  and  Loss 237—239 

Insurance 239—241 

Interest 241—247 

Partial  Payments 247—251 

Compound  Interest 251—253 

Discount 253—255 

Bank  Discount 255—257 

Equation  of  Payments 257 — 260 

Assessing  Taxes 260 — 263 

Coins  and  Currency 263 — 264 

Reduction  of  Currencies 264 — 265 

Exchange , 265—268 

Duties 268—271 

Alligation  Medial 271—272 

Alligation  Alternate 272—276 


VIII  CONTENTS. 

INVOLUTION. 

Definition  of,  &c '."...  276 

EVOLUTION. 

Definition  of,  &c 277 

Extraction  of  the  Square  Root 277 — 282 

Applications  in  Square  Root 282 — 285 

Extraction  of  the  Cube  Root 285—289 

Applications  in  Cube  Root 289 — 290 

ARITHMETICAL  PROGRESSION. 

Definition  of,  &c. , 290—291 

Different  Cases 291—294 

GEOMETRICAL  PROGRESSION. 

Definition  of,  &c 294—295 

Cases 295—297 

PROMISCUOUS  QUESTIONS. 

Questions  for  Practice 298—303 

MENSURATION. 

To  find  the  area  of  a  Triangle S03 

To  find  the  area  of  a  Square,  Rectangle,  &c 303 

To  find  the  area  of  a  Trapezoid 304 

To  find  the  circumference  and  diameter  of  a  Circle 304 

To  find  the  area  of  a  Circle 305 

To  find  the  surface  of  a  Sphere 305 

To  find  the  contents  of  a  Sphere 305 

To  find  the  convex  surface  of  a  Prism 306 

To  find  the  contents  of  a  Prism 306 

To  find  the  convex  surface  of  a  Cylinder , 307 

To  find  the  contents  of  a  Cylinder 

To  find  the  contents  of  a  Pyramid 

To  find  the  contents  of  a  Cone 308 

GAUGING. 

Rules  for  Gauging 309 

APPENDIX. 

Forms  relating  to  Business  in  General ,  310—813 


ARITHMETIC 


DEFINITIONS. 

1.  A  SINGLE  THING  is  called  one  or  a  unit. 

2.  A  NUMBER  is  a  unit,  or  a  collection  of  units.     The  unit 
is  called  the  base  of  the  collection.     The  primary  base  of 
every  number  is  the  unit  one. 

3.  Each  of  the  words,  or  terms,  one,  two,  three,  four,  &c., 
denotes  how  many  things  are  taken.     These  terms  are  gene- 
rally called  numbers ;    though,  in  fact,  they  are   but   the 
names  of  numbers. 

4.  The  term,  one,  has  no  reference  to  the  kind  of  thing  to 
which  it  is  applied  :  and  is  called  an  abstract  unit. 

5.  An  abstract  number  is  one  whose  unit  is  abstract :  thus, 
three,  four,  six,  &c.,  are  abstract  numbers. 

6.  The  term,  one  foot,  refers  to  a  single  foot,  and  is  called 
a  denominate  unit :  hence, 

7.  A  denominate  number  is  one  whose  unit  is  named,  or 
denominated :  thus,  three  feet,  four  dollars,  five  pounds,  are 
denominate  numbers.     These  numbers  are  also  called  con- 
crete numbers. 


L  "What  is  a  single  thing  called  ? 

2.  What  is  a  number  V  What  is  the  unit  called  ?  What  is  the 
primary  base  of  every  number  ? 

a  What  does  each  of  the  words,  one,  two,  three,  denote  ?  What  are 
these  words  generally  called  ?  WThat  are  they,  in  fact '? 

4.  Has  the  term  one  any  reference  to  the  thing  to  which  it  may  be 
applied  ?     What  is  it  called  ? 

5.  What  is  an  abstract  number?    Give  examples  of  abstract  num- 
bers. 

6.  What  does  the  term  one  foot  refer  to  ?    What  is  it  called  ? 

7.  What  is  a  denominate  number  ?  Give  examples  of  denominate  num- 
bers.   What  are  denominate  numbers,  also  called  ? 


10  DEFINITIONS. 

8.  A  SIMPLE  NUMBER  is  a  single  collection  of  units. 

9.  QUANTITY  is  any  thing  which  can  be  increased,  dimin- 
ished and  measured. 

10.  SCIENCE  treats  of  the  properties  and  relations  of  things  : 
ART  is  the  practical  application  of  the  principles  of  Science. 

11.  ARITHMETIC  treats  of  numbers.     It  is  a  science  when 
it  makes  known  the  properties  and  relations  of  numbers  ;  and 
an  art,  when  it  applies  principles  of  science  to  practical  pur- 
poses. 

12.  A  PROPOSITION  is  something  to  be  done,  or  demonstrated. 

13.  An  ANALYSIS  is  an  examination  of  the  separate  parts 
of  a  proposition. 

14.  An   OPERATION  is  the  act  of  doing  something  with 
numbers.     The  number  obtained  by  an  operation  is  called  a 
result,  or  answer. 

15.  A  RULE  is  a  direction  for  performing  an  operation,  and 
may  be  deduced  either  from  an  analysis  or  a  demonstration. 

1C.  There  are  five  fundamental  processes  of  Arithmetic : 
Notation  and  Numeration,  Addition,  Subtraction,  Multiplica- 
tion and  Division. 

EXPRESSING  NUMBERS. 

17.  There  are  three  methods  of  expressing  numbers : 

1st.  By  words,  or  common  language  ; 

2d.  By  capital  letters,  called  the  Roman  method  ; 

3d.  By  figures,  called  the  Arabic  method. 


8.  What  is  a  simple  number  ? 

9.  What  is  quantity  ? 

10.  Of  what  does  Science  treat  ?    What  is  Art  ? 

11.  Of  what  does  Arithmetic  treat?    When  is  it  a  science?    When 
an  art  ? 

12.  What  is  a  Proposition  ? 

13.  What  is  an  Analysis  ? 

14.  What  is  an  Operation  ?    What  is  the  number  obtained  called  ? 

15.  WThat  is  a  Rule  ?    How  may  it  be  deduced  ? 

16.  How  many  fundamental  rules  are  there  ?    What  are  they  ? 

17.  How  many  methods  are  there  of  expressing  numbers?    What 
are  they  ? 


NOTATION.  11 

BY  WORDS. 

18.  A  single  thing  is  called  -  One. 
One      and  one  more       -  Two. 
Two     and  one  more       -  Three. 
Three  and  one  more       -  Four. 
Four    and  one  more      -  .Five. 
Five     and  one  more      -  Six. 
Six       and  one  more       -  Seven. 
Seven  and  one  more '     -  Eight. 
Eight  and  one  more       -  Nine. 
Nine    and  one  more      -  Ten. 
&c.                             &c.  &c. 

Each  of  the  words,  one,  two,  three,  four,  Jive,  six,  &c., 
denotes  how  many  things  are  taken  in  the  collection. 

NOTATION. 

19.  NOTATION  is  the  method  of  expressing  numbers  either 
by  letters  or  figures.     The  method  by  letters,  is  called  Roman 
Notation;  the  method  by  figures  is  called  Arabic  Notation. 

ROMAN  NOTATION. 

20.  In  the  Roman  Notation,  seven  capital  letters  are  used, 
viz  :  I,  stands  for  one  ;  V,  hv  five  ;  X,  for  ten;  L,  for  fifty  ; 
C,  for  one  hundred  ;  D,  for  five  hundred',  and  M,  for  one 
thousand.     All  other  numbers  are  expressed  by  combining 
the  letters  according  to  the  following 


ROMAN  TABLE. 


I.  -    -    -    -  One. 

II.  -    -    -    -  Two. 

III.  -    -    -  Three. 

IV.  ...  Four. 

V.  .-.-  Five. 

VI.  ...  Six. 

VII.  -    -     -  Seven. 

VIII.  -    -    -  Eight. 

IX.  ---  Nine. 

X.  -    ---  Ten. 
XX.    -    -    -  Twenty. 
XXX.-    -    -  Thirty. 
XL.    ---  Forty. 
L.    -    -        -  Fifty. 
LX.     -    -    -  Sixty. 


LXX.       -  .  Seventy. 

LXXX.    -  -  Eighty. 

XC.     -    -  -  Ninety. 

£.----  One  hundred. 

CC.     ---  Two  hundred. 

CCC.  -    -  -  Three  hundred. 

CCCC.      -  -  Four  hundred. 

D.  -    -    -  -  Five  hundred. 

DC.     -    -  -  Six  hundred. 

DCC.  -    -  -  Seven  hundred. 

DCCC.     -  -  Eight  hundred. 

DCCCC.  .  -  Nine  hundred. 

M.  -     -    -  -  One  thousand. 

MD.    -    -  -  Fifteen  hundred. 

MM.    -    -  -  Two  thousand. 


12  NOTATION. 

NOTE. — The  principles  of  this  Notation  are  these : 

1.  Every  time  a  letter  is  repeated,  the  number  which  it  denotes 
is  also  repeated. 

2.  If  a  letter  denoting  a  less  number  is  written  on  the  right  of 
one  denoting  a  greater,  their  sum  will  be  the  number  expressed. 

3.  If  a  letter  denoting  a  less  number  is  written  on  the  left  of 
one  denoting  a  greater,  their  difference  will  be  the  number  ex- 
pressed. 

EXAMPLES    IN    ROMAN    NOTATION. 

Express  the  following  numbers  by  letters  : 

1.  Eleven. 

2.  Fifteen. 

3.  Nineteen. 

4.  Twenty-nine. 

5.  Thirty-five. 

6.  Forty-seven. 
7'.  Ninety-nine. 

8.  One  hundred  and  sixty. 

9.  Four  hundred  and  forty-one, 

10.  Five  hundred  and  sixty-nine. 

11.  One  thousand  one  hundred  and  six, 

12.  Two  thousand  and  twenty-five. 

13.  Six  hundred  and  ninety-nine. 

14.  One  thousand  nine  hundred  and  twenty-five. 

15.  Two  thousand  six  hundred  and  eighty. 

16.  Four  thousand  nine  hundred  and  sixty-five. 
It.  Two  thousand  seven  hundred  and  ninety-one. 

18.  One  thousand  nine  hundred  and  sixteen. 

19.  Two  thousand  six  hundred  and  forty-one. 

20.  One  thousand  three  hundred  and  forty-two. 


19.  What  is  Notation  ?  What  is  the  method  by  letters  called  ?  What 
is  the  method  by  figures  called  ?  • 

30.  How  many  letters. are  used  in  the  Roman  notation?  Which  are 
they  ?  What  does  each  stand  for  ? 

NOTE. — What  takes  place  when  a  letter  is  repeated  ?  If  a  letter  de- 
noting a  less  number  be  placed  on  the  right  of  one  denoting  a  greater, 
how  are  they  read  ?  If  the  letter  denoting  the  less  number  be  written 
on  the  left,  how  are  they  read  ? 

21.  What  is  Arabic  Notation  ?  How  many  figures  are  used?  What 
do  they  form?  Name  the  figures.  How  many  things  does  1  express  ? 
How  many  things  does  2  express  ?  How  many  units  in  3?  In  4  ?  In 
6  ?  In  9  ?  In  8  ?  What  docs  0  express  ?  What  are  the  other  figures 
called? 


NOTATION.  13 

ARABIC  NOTATION. 

21.  Arabic  Notation  is  the  method  of  expressing  numbers 
by  figures.     Ten  figures  are  used,  and  they  form  the  alphabet 
of  the  Arabic  Notation. 

They  are    0  called  zero,  cipher,  or  Naught. 

1  One. 

2  Two. 

3  Three. 

4  Four. 

5  -  Five. 

6  -  -  Six. 

7  Seven. 

8  -  Eight. 

9  -  -  Nine. 

1  expresses  a  single  thing,  or  the  unit  of  a  number. 

2  two  things  or  two  units. 

3  three  things  or  three  units. 

4  four  things  or  four  units. 

5  five  things  or  five  units. 

6  six  things  or  six  units. 

7  seven  things  or  seven  units. 

8  eight  things  or  eight  units. 

9  nine  things  or  nine  units. 

The  cipher,  0,  is  used  to  denote  the  absence  of  a  thing : 
Thus,  to  express  that  there  are  no  apples  in  a  basket,  we 
write  the  number  of  apples  is  0.  The  nine  other  figures  are 
called  significant  figures,  or  Digits. 

22.  We  have  no  single  figure  for  the  number  ten.     We 
therefore  combine  the  figures  already  known.     This  we  do  by 
writing  0  on  the  right  hand  of  1,  thus  : 

10,  which  is  read  ten. 

This  10  is  equal  to  ten  of  the  units  expressed  by  1.  It  is, 
however,  but  a  single  ten,  and  may  be  regarded  as  a  unit, 
the  value  of  which  is  ten  times  as  great  as  the  unit  1.  It  is 
called  a  unit  of  the  second  order. 

22.  Have  we  a  separate  character  for  ten  ?  How  do  we  express  ten  ? 
To  how  many  units  1  is  ten  equal  ?  May  we  consider  it  a  single  unit  ? 
Of  what  order  ? 


14  NOTATION. 

23.  When  two  figures  are  written  by  the  side  of  each  other, 
the  one  on  the  right  is  in  the  place  of  units,  and  the  other  in 
the  place  of  tens,  or  of  units  of  the  second  order.     Each  unit 
of  the  second  order  is  equal  to  ten  units  of  the  first  order. 

When  units  simply  are  named,  units  of  the  first  order  are 
always  meant. 

Two  tens,  or  two  units  of  the  second  order,  are  written  20 

Three  tens,  or  three  units  of  the  second  order,  are  written  3Q 

Four  tens,  or  four  units  of  the  second  order,  are  written  40 

Five  tens,  or  five  units  of  the  second  order,  are  written  50 

Six  tens,  or  six  units  of  the  second  order,  are  written  (50 

Seven  tens,  or  seven  units  of  the  second  order,  are  written  *JQ 

Eight  tens,  or  eight  units  of  the  second  order,  are  written  gQ 

Nine  tens,  or  nine  units  of  the  second  order,  are  written  99 

These  figures  are  read,  twenty,  thirty,  forty,  fifty,  sixty, 
"seventy,  eighty,  ninety. 

The  intermediate  numbers  between  10  and  20,  between  20 

and  30,  &c.,  may  be  readily  expressed  by  considering  their 

tens  and  units.     For  example,  the  number  twelve  is  made 

up  of  one  ten  and  two  units.     It  must  therefore  be  written 

by  setting  1  in  the  place  of  tens,  and  2  in  the  place  of  units  : 

thus,     -  12 

Eighteen  has  1  ten  and  8  units,  and  is  written  -        Jg 

Twenty-five  has  2  tens  and  5  units,  and  is  written    -  -        25 

Thirty-seven  has  3  tens  and  7  units,  and  is  written  -  3*7 

Fifty-four  has  5  tens  and  4  units,  and  is  written     "  -  -        54 

Hence,  any  number  greater  than  nine,  and  less  than  one 
hundred,  may  be  expressed  by  two  figures. 

24.  In  order  to  express  ten-units  of  the  second  order,  or 
one  hundred,  we  form  a  new  combination. 

It  is  done  thus,  •      .      -         100 

by  writing  two  ciphers  on  the  right  of  1.     This  number  is 
read,  one  hundred. 

23.  When  two  figures  are  written  by  the  side  of  each  other,  what  is 
the  place  on  the  right  called?  The  place  on  the  left?  When  units 
simply  are  named,  what  units  are  meant  ?  How  many  units  of  the 
second  order  in  20?  In  80?  In  40?  In  50?  In  60?  In  70?  In 
80  ?  In  90  ?  Of  what  is  the  number  12  made  up  ?  Also  18,  25,  37, 
54  ?  What  numbers  may  be  exprsesed  by  two  figures  ? 


NOTATION.  15 

Now  this  one  hundred  expresses  10  units  of  the  second 
order,  or  100  units  of  the  first  order.  The  one  hundred  is  but 
an  individual  hundred,  and,  in  this  light,  may  be  regarded 
as  a  unit  of  the  third  order. 

We  can  now  express  any  number  less  than  one  thousand. 

For  example,   in    the    number  three    hundred   and  . 

seventy-five,  there  are  5  units,  7  tens,  and  3  hundreds,  c    g    .-§ 

Write,  therefore,  5  units  of  the  first  order,  7  units  of  the  Jj    %    § 

second  order,  and  3  of  the  third  *    and  read  from  the  375 
right,  units,  tens,  hundreds. 

In  the  number  eight  hundred  and  ninety-nine,  there  w  K-  _» 
are  9  units  of  the  first  order,  9  of  the  second,  and  8  of  &  §  3 

the  third ;  ar«d  is  read,  units,  tens,  hundreds.  •**    * 

o    y    y 

In  the  number  four  hundred  and  six,  there  are  6  units  »  .  & 
of  the  first  order,  0  of  the  second,  and  4  of  the  third. 

The  right  hand  figure  always  expresses  units  of    4    ' 
the  first  order  ;  the  second,  units  of  the  second  order  ;  and 
the  third,  units  of  the  third  order. 

25.  To  express  ten  units  of  the  third  order,  or  one  thous- 
and, we  form  a  new  combination  by  writing  three  ciphers  on 
the  right  of  1  ;  thus,  1000 

Now,  this  is  but  one  single  thousand,  and  may  be  regarded 
as  a  unit  of  the  fourth  order. 

Thus,  we  may  form  as  many  orders  of  units  as  we  please  : 

a  single  unit  of  the  first  order  is  expressed  by  1 , 

a  unit  of  the  second  order  by  1  and  0  ;  thus,  10, 

a  unit  of  the  third  order  by  1  and  two  O's  ;  100, 

a  unit  of  the  fourth  order  by  1  and  three  O's  ;  1000, 

a  unit  of  the  fifth  order  by  1  and  four  O's  ;  10000  ; 
and  so  on,  for  units  of  higher  orders  : 


24.  How  do  you  write  one  hundred?  To  how  many  units  of  the 
second  order  is  it  equal  ?  To  how  many  of  the  lirst  order  ?  May  it  be 
considered  a  single  unit  ?  Of  what  order  is  it  ?  How  many  units  of 
the  third  order  in  200?  In  300?  In  400?  In  500?  In  600?  Of 
what  is  the  number  375  composed  ?  The  number  899  ?  The  number 
406  ?  What  numbers  may  be  expressed  by  three  figures  ?  What 
order  of  units  will  each  figure  express  ? 


16  NOTATION. 

26.  Therefore, 

1st.  The  same  figure  expresses  different  units  according 
to  the  place  which  it  occupies : 

2d.  Units  of  the  first  order  occupy  the  place  on  the  right ; 
units  of  the  second  order,  the  second  place  ;  units  of  the  third 
order,  the  third  place  ;  and  so  on  for  places  still  to  the  left : 

3d.  Ten  units  of  the  first  order  make  one  of  the  second  ; 
ten  of  the  second,  one  of  the  third  ;  ten  of  the  third,  one  of 
the  fourth  ;  and  so  on  for  the  higher  orders : 

4th.  When  figures  are  written  by  the  side  of  each  other, 
ten  units  in  any  one  place  make  one  unit  of  the  place  next 
to  the  left. 

EXAMPLES    IN    WRITING    THE    ORDERS    OF    UNITS. 

1.  Write  3  tens. 

2.  Write  8  units  of  the  second  order. 

3.  Write  9  units  of  the  first  order. 

4.  Write  4  units  of  the  first  order,  5  of  the  second,  6  of  the 
third,  and  8  of  the  fourth. 

5.  Write  9  units  of  the  fifth  order,  none  of  the  fourth,  8  of 
the  third,  7  of  the  second,  and  6  of  the  first.        Ans.  90876. 

6.  Write  one  unit  of  the  sixth  order,  5  of  the  fifth,  4  of  the 
fourth,  9  of  the  third,  7  of  the  second,  and  0  of  the  first. 

Ans. 

7.  Write  4  units  of  the  eleventh  order. 

8.  Write  forty  units  of  the  second  order. 

9.  Write  60  units  of  the  third  order,  with  four  of  the  2d, 
and  5  of  the  first. 

10.  Write   6  units  of  the  4th  order,  with  8  of  the  3d, 
4  of  the  1st. 

25.  To  what  are  ten  units  of  the  third  order  equal  ?    How  do  you 
write  it?    How  is  a  single  unit  of  the  first  order  written  ?    How  do 
you  write  a  unit  of  the  second  order  ?    One  of  the  third  ?    One  of  the 
fourth  ?    One  of  the  fifth  ? 

26.  On  what  does  the  unit  of  a  figure  depend  ?    What  is  the  unit  of 
the  first  place  on  the  right  ?    What  is  the  unit  of  the  second  place  ? 
What  is  the  unit  of  the  third  place  ?    Of  the  fourth  ?    Of  the  fifth  ? 
Sixth  ?    How  many  units  of  the  first  order  make  one  of  the  second  ? 
How  many  of  the  second  one  of  the  third  ?    How  many  of  the  third  one 
of  the  fourth,  &c.     When  figures  are  written  by  the  side  of  each  other, 
how  many  units  of  any  place  make  one  unit  of  the  place  next  to  the 
left? 


NUMERATION.  17 

11.  Write  9  units  of  the  5th  order,  0  of  the  4th,  8  of  the 
3d,  1  of  the  2d,  and  3  of  the  1st. 

12.  Write  7  units  of  the  6th  order,  8  of  the  5th,  0  of  the 
4th,  5  of  the  3d,  7  of  the  2d,  and  1  of  the  llth. 

13.  Write  9  units  of  the  7th  order,  0  of  the  6th,  2  of  the 
5th,  3  of  the  4th,  9  of  the  3d,  2  of  the  2d,  and  9  of  the  1st. 

14.  Write  8  units  of  the  8th  order,  6  of  the  7th,  9  of  the 
6th,  8  of  the  5th,  1  of  the  4th,  0  of  the  3d,  2  of  the  2d,  and 
8  of  the  1st. 

15.  Write  1  unit  of  the  9th  order,  6  of  the  8th,  9  of  the 
7th,  7  of  the  6th,  6  of  the  5th,  5  of  the  4th,  4  of  the  3d,  3  of 
the  2d,  and  2  of  the  1st. 

16.  Write  8  units  of  the  10th  order,  0  of  the  9th,  0  of  the 
8th,  0  of  the  7th,  9  of  the  6th,  8«of  the  5th,  0  of  the  4th, 
3  of  the  3d,  2  of  the  2d,  and  0  of  the  1st. 

17.  Write  7  units  of  the  ninth  order,  with  6  of  the  7th,  9 
of  the  third,  8  of  the  2d,  and  9  of  the  1st. 

18.  Write  6  units  of  8th  order,  with  9  of  the  6th,  4  of  the 
5th,  2  of  the  3d,  and  1  of  the  1st. 

19.  Write  14  units  of  the  12th  order,  with  9  of  the  10th, 
6  of  the  8th,  7  of  the  6th,  6  of  the  5th,  5  of  the  3d,  and  3 
of  the  first. 

20.  Write  13  units  of  the  13th  order,  8  of  the  12th,  7  of 
the  9th,  6  of  the  8th,  9  of  the  7th,  7  of  the  6th,  3  of  the  4th, 
and  9  of  the  first. 

21.  Write  9  units  of  the  18th  order,  7  of  the  16th,  4  of  the 
loth,  8  of  the  12th,  3  of  the  llth,  2  of  the  10th,  1  of  the  9th, 
0  of  the  8th,  6  of  the  7th,  2  of  the  third,  and  1  of  the  1st. 

NUMERATION. 

27.  NUMERATION  is  the  art  of  reading  correctly  any  num- 
ber expressed  by  figures  or  letters. 

The  pupil  has  already  been  taught  to  read  all  numbers  from 
one  to  one  thousand.  The  Numeration  Table  will  teach  him 
to  read  any  number  whatever ;  or,  to  express  numbers  in  words. 


27.  What  is  Numeration?  What  is  the  unit  of  the  first  period? 
What  is  the  unit  of  the  second  ?  Of  the  third  ?  Of  the  fourth  ?  Of 
the  fifth?  Sixth?  Seventh?  Eighth?  Give  the  rale  for  reading 
numbers. 


NUMERATION. 


NUMERATION  TABLE. 


6th  Period,    5th  Period.    4th  Period.    3d  Period,    2d  Period.    1st  Period. 
Quadrillions.    Trillions.        Billions.         Millions.    Thousands.        Units. 


II;    I ! !    I ! !   I ! !    l-s : 

ip  .      ?«.      §  *  !     ^  8  •     -^1  i 

S3        -§25        |||       |||       |a|       | 


, 

• 

6, 
8  2, 

6, 
7  5, 
879, 
023, 
301, 

„ 

, 

. 

. 

123, 

087, 

7, 

000, 

735, 

B 

. 

. 

4  3, 

2  1  0, 

460, 

548, 

000, 

087, 

(. 

. 

6, 

245, 

289, 

421, 

7  2, 

549, 

1  3  6, 

822, 

» 

894, 

602, 

043, 

288, 

7, 

641, 

000, 

907, 

456, 

• 

8  4, 

912, 

876, 

4  1  9, 

285, 

912, 

761, 

257, 

327, 

826, 

6, 

407, 

2  1  2, 

936, 

876, 

541, 

5  7, 

289, 

678, 

541, 

297, 

313, 

920, 

323, 

842, 

768, 

319, 

675, 

NOTES. — 1.  Numbers  expressed  by  more  than  three  figures  are 
•written  and  read  by  periods,  as  shown  in  the  above  table. 

2.  Each  period  always  contains  three  figures,  except  the  last, 
which  may  contain  either  one,  two,  or  three  figures. 

3.  The  unit  of  the  first,  or  right-hand  period,  is  1  ;  of  the  second 
period,  1  thousand ;  of  the  3d,  1  million ;  of  the  fourth,  1  billion ; 
and  so,  for  periods,  still  to  the  left. 

4.  To  quadrillions  succeed   quintillions,  sextillions,   septillions, 
octillions,  &c. 

5.  The  pupil  should  be  required   to  commit,  thoroughly,   the 
names  of  the  periods,  so  as  to  repeat  them  in  their  regular  order 
from  left  to  right,  as  well  as  from  right  to  left. 


NUMERATION. 


19 


RULE  FOR  READING  NUMBERS. 

I.  Divide  the  number  into  periods  of  three  figures  each, 
beginning  at  the  right  hand. 

II.  Name  the  order  of  each  figure,  beginning  at  the  right 
hand. 

III.  Then,  beginning  at  the  left  hand,  read  each  period  an 
if  it  stood  alone,  naming  its  unit. 


EXAMPLES    IN    READING    NUMBERS. 

28.  Let  the  pupil  point  off  and  read  the  following  numbers 
-then  write  them  in  words. 


19. 
20. 
21. 
22. 


67 

125 

6256 

4697 

23697 

412304 


7. 

8. 

9. 
10. 
11. 
12. 


6124076 
8073405 
26940123 
9602316 
87000032 
1987004086 

13. 

14. 
15. 
16. 
17. 

18. 

804321049 
90067236708 
870432697082 
1704291672301 
3409672103604 
49701342641714 

8760218760541 

904326170365 

30267821040291 

907620380467026 


23.  9080620359704567 

24.  9806071234560078 

25.  30621890367081263 

26.  350673123051672607 


NOTE. — Let  each  of  the  above  examples,  after  being  written  on 
the  black  board,  be  analyzed  as  a  class  exercise ;  thus : 

Ex.  1.  How  many  tens  in  67  ?    How  many  units  over  ? 

2.  In  125,  how  many  hundreds  in  the  hundreds  place?    How 
many  tens  in  the   tens   place  ?    How  many  units  in  the  units 
place  ?    How  many  tens  in  the  number  ? 

3.  In  6256,  how  many  thousands  in  the  thousands  place  ?    How 
many  hundreds  in  the  hundreds  place  ?    How  many  tens  in  the 
tens  place  ?     How  many  units  in  the  units  place  ? 

4.  How  many    thousands  in   the  number  4697?     How  many 
hundreds  ?     How  many  tens  ?     How  many  units  ? 

5.  How  many  thousands  in  the  number  23697?     How  many 
hundreds  ?     How  many  tens  ?    How  many  units  ? 

6.  How  many  hundreds  of  thousands  in  412304?    How  many 
ten    thousands  ?    How  many  thousands  ?    How  many  hundreds  ? 
How  many  tens  ?    How  many  units  ? 


28.  Name  the  units  of  each  order  in  example  9  ?    In  10  ?    In  15  ? 
In  30  ?    Give  the  rule  for  writing  numbers. 


20  NUMERATION. 


RULE    FOR   WRITING    NUMBERS,    OR    NOTATION. 

I.  Begin  at  the  left  hand  and  write  each  period  in  order,  as 
if  it  icere  a  period  of  units. 

II.  When  the  number  of  any  period,  except  the  left  hand 
period,  is  expressed  by  less  than  three  figures,  prefix  one  or  two 
ciphers  ;  and  when  a  vacant  period  occurs,  fill  it  with  ciphers. 


EXAMPLES    IX    NOTATION. 

29.  Express  the  following  numbers  in  figures  : 

1.  One  hundred  arid  five. 

2.i  Three  hundred  and  two. 

3.  Five  hundred  and  nineteen. 

_.  4.  One  thousand  and  four. 

5.  Eight  thousand  seven  hundred  and  one. 

6.  Forty  thousand  four  hundred  and  six.  / 

7.  Fifty-eight  thousand  and  sixty-one. 

8.  Ninety-nine  thousand  nine  hundred  and  ninety-nine. 

9.  Four  hundred  and  six  thousand  and  forty-nine. 

10.  Six  hundred  and  forty-one  thousand,  seven  hundred 
and  twenty-one. 

11.  One  million,  four  hundred  and  twenty-one  thousands, 
six  hundred  and  two. 

12.  Nine  millions,  six  hundred  and  twenty-one  thousands, 
and  sixteen.       /  ~j£ 

13.  Ninety-four  millions,  eight  hundred  and  seven  thous- 
ands, four  hundred  and  nine. 

14.  Four  billions,  three  hundred  and  six  thousands,  nine 
hundred  and  nine. 

15.  Forty-nine  billions,  nine  hundred  and  forty-nine  thous- 
ands, and  sixty-five. 

16.  Nine  hundred  and  ninety  billions,  nine  hundred  and 
ninety-nine  millions,  nine  hundred  and  ninety  thousands,  nine 
hundred  and  ninety-nine. 

17.  Four  hundred  and  nine  billions,  two  hundred  and  nine 
thousands,  one  hundred  and  six. 

18.  Six  hundred  and  forty-five  billions,  two  hundred  and 
sixty-nine  millions,  eight  hundred  and  fifty-nine  thousands, 
nine  hundred  and  six. 


NUMERATION.  iJl 

19.  Forty-seven  millions,  two  hundred  and  four  thousands, 
eight  hundred  and  fifty-one. 

20.  Six   quadrillions,  forty-nine  trillions,  seventy-two  bil- 
lions, four  hundred  and  seven  thousands,  eight  hundred  and 
sixty-one. 

21.  Eight  hundred  and  ninety-nine  quadrillions,  four  hun- 
dred and  sixty  trillions,  eight  hundred  and  fifty  billions,  two 
hundred  millions,  five  hundred  and  six  thousands,  four  hun- 
dred and  ninety-nine. 

22.  Fifty-nine  trillions,  fifty-nine  billions,  fifty-nine  millions, 
fifty-nine  thousands,  nine  hundred  and  fifty-nine. 

23.  Eleven  thousands,  eleven  hundred  and  eleven. 

24.  Nine  billions  and  sixty-five. 

25.  Write  three*  hundred  and  four  trillions,  one  million, 
three  hundred  and  twentv-one  thousands,  nine  hundred  and 
forty-one. 

26.  Write  nine  trillions,  six  hundred  and  forty  billions, 
with  7  units  of  the  ninth  order,  6  of  the  seventh  order,  8  of 
the  fifth,  2  of  the  third,  1  of  the  second,  and  3  of  the  first. 

27.  Write  three  hundred  and  five  trillions,  one  hundred 
and  four  billions,  one  million,  with  4  units  of  the  fifth  order, 
5  of  the  fourth,  7  of  the  second,  and  4  of  the  first. 

28.  Write  three  hundred  and  one  billions,  six  millions,  four 
thousands,  with  8  units  of  the  fourteenth  order.  6  of  the 
third,  and  two  of  the  second. 

29.  Write  nine  hundred  and  four  trillions  six  hundred  and 
six,  with  4  units  of  the  eighteenth  order,  five  of  the  sixteenth, 
four  of  the  twelfth,  seven  of  the  ninth,  and  6  of  the  fifth. 

30.  Write  sixty-seven  quadrillions,  six  hundred  and  forty- 
one  billions,  eight  hundred  and  four  millions,  six  hundred  and 
forty-four. 

31.  Write  eight  hundred  and  three  quintillions,  sixty-nine 
billions,  four  hundred  and  forty  millions,  nine  hundred  thous- 
and and  three. 

32.  Write  one  hundred  and  fifty-nine  sextillions,  four  hun- 
dred and  five  billions,  two  hundred  and  one  millions,  three 
thousand  and  six. 

33.  Write  four  hundred  and  four  septillions,  nine  hundred 
and  three  sextillions,  two  hundred  and  one  quintillions,  forty 
quadrillions,  and  three  hundred  and  four. 


ADDITION. 


ADDITION. 

30.  1.  John  has  two  apples  and  Charles  has  three :  how 
many  have  both  ? 

ANALYSIS. — If  John's  apples  be  placed  with  Charles's,  there  will 
be  five  apples. 

The  operation  of  finding  how  many  apples  both  have  is  called 
Addition. 

ADDITION  TABLE. 


2  and    0  are    2 

3  and    0  are    3 

4  and    0  are    4 

5  and    0  are    5 

2  and    1  are    3 

3  and    1  are    4 

4  and    1  are    5 

5  and    1  are    G 

2  and    2  are    4 

3  and    2  are    5 

4  and    2  are    G 

5  and    2  are    V 

2  and    3  are    5 

3  and    3  are    G 

4  and    3  are    7 

5  and    3  are    8 

2  and    4  are    6 

3  and    4  are    7 

4  and    4  are    8 

5  and    4  are    9 

2  and    5  are    7 

3  and    5  are    8 

4  and    5  are    9  5  and    5  are  10 

2  and    6  are    8 

3  and    6  are    9 

4  and    6  are  10 

5  and    6  are  1  1 

2  and    7  are    9 

3  and    7  are  10 

4  and    7  are  11 

5  and    7  are  12 

2  and    8  are  10 

3  and    8  are  11 

4  and    8  are  12 

5  and    8  are  ]3 

2  and    9  are  1  1 

3  and    9  are  12 

4  and    9  are  13 

5  and    9  are  14 

2  and  10  are  12 

3  and  10  are  13 

4  and  10  are  14 

5  and  10  are  15 

6  and    0  are    6 

7  and    0  are    7 

8  and    0  are    8 

9  and    0  are    9 

6  and    1  are    7 

7  and    1  are    8 

8  and    1  are    9 

9  and    1  are  10 

G  and    2  are    8 

7  and    2  are    9 

8  and    2  are  10 

9  and    2  are  11 

G  and    3  are    9 

7  and    3  are  10 

8  and    3  are  11 

9  and    3  are  12 

6  and    4  are  10 

7  and    4  are  11 

8  and    4  are  12 

9  and    4  are  13 

G  and    5  are  11 

7  and    5  are  12 

8  and    5  are  13 

9  and    5  are  14 

6  and    6  are  12 

7  and    G  are  13 

8  and    6  are  14 

9  and    6  are  15 

6  and    7  are  13 

7  and    7  are  14 

8  and    7  are  15 

9  and    7  are  16 

G  and    8  are  14 

7  and    8  are  15 

8  and    8  are  16 

9  and    8  are  17 

6  and    9  are  15 

7  and    9  are  16 

8  and    9  are  17 

9  and    9  are  18 

6  and  10  are  16 

7  and  10  are  17 

8  and  10  are  18 

9  and  10  are  19 

2.  James  has  5  marbles  and  William  7  ?  how  many  have 
both? 

3.  Mary  has  6  pins  and  Jane  9  :  how  many  have  both  ? 

4.  How  many  are  4  and  5  and  3  ? 

5.  How  many  are  6  and  4  and  9  ? 

6.  How  many  are  3  and  7  ?  4  and  6  ?  2  and  8  ?  5  and  5  ? 
9  and  1?  10  arid  0  ?  0  and  10? 

7.  How  many  are  6  and  3  and  9  ?     How  many  are  18  and 
2?  18  and  3?  18  and  5? 


SIMPLE  NUMBERS.  23 

8.  James  had  9  cents  and  Henry  gave  him  eight  more : 
how  many  had  he  in  all  ? 

PRINCIPLES    AND    EXAMPLES. 

31.  James  has  3  apples  and  John  4  :    how  many  have 
both  ?     Seven  is  called  the  sum  of  the  numbers  3  and  4. 

The  SUM  of  two  or  more  numbers  is  a  number  which  con- 
tains as  many  units  as  all  the  numbers  taken  together. 

ADDITION  is  the  operation  of  finding  the  sum  of  two  or 
more  numbers. 

OF   THE    SIGNS. 

32.  The   sign    +    is   called   plus,   which   signifies  more. 
When  placed  between  two  numbers  it  denotes  that  they  are 
to  be  added  together. 

The  sign  =  is  called  the  sign  of  equality.  When  placed 
between  two  numbers  it  denotes  that  they  are  equal ; 
that  is,  that  they  contain  the  same  number  of  units.  Thus  : 
3  +  2  =  5 

2+3=     how  many? 

1+2  +  4=     how  many  ? 

2  +  3  +  5  +  1=     how  many? 

6  +  7+2+3=     how  many? 

1  +  6  +  7+2  +  3=     how  many? 

1+2+3+4  +  5  +  6  +  7+8  +  9=     how  many? 

1.  James  has  14  cents,  and  John  gives  him  21 :  how  many 
will  he  then  have  ? 

OPERATION. 

14 

ANALYSIS. — Having  written  the  numbers,  as  at  the     21 
right  of  the  page,  draw  a  line  beneath  them. 

oO  cents. 

The  first  number  contains  four  units  and  1  ten,  the  second  1 
unit  and  two  tens.  We  write  the  units  in  one  column  and  the 
tens  in  the  column  of  tens. 


31.  What  is  the  sum  of  two  or  more  numbers?    What  is  addition ? 

32.  What  is  the  sign  of  addition  ?    What  is  it  called  ?    What  does 
it  signify?    Express  the  sign  of  equality?    When  placed  between  two 
numbers  what  does  it  show  ?    When  is  a  number  equal  to  the  sum 
of  other  numbers  ?    Give  an  example. 


24:  ADDITION. 

We  then  begin  at  the  right  hand,  and  say  1  and  4  are  5,  which 
we  set  down  below  the  line  in  the  units'  place.  We  then  add 
the  tens,  and  write  the  sum  in  the  tens'  place.  Hence,  the  sum 
is  3  tens  and  5  units,  or  35  cents. 

OPERATION. 

24 

2.  John  has  24  cents,  and  William  62  :  how         62 
many  have  both  of  them  ?  gg 

OPERATION. 

3.  A  farmer  has  160  sheep  in  one  field,  20  in     1^ 
another,  and  16  in  another  :  how  many  has  he 

in  all  ? 

196 

OPERATION. 

4.  What  is  the  sum  of  328  and  111  ?  ® 


499 

(5.)  (6.)  (7.)  (8.) 
427  329  3034  8094 
242  260  6525  1602 
330  100  236  103 


999 
9.  What  is  the  sum  of  304  and  273  ? 

10.  What  is  the  sum  of  3607  and  4082  ? 

11.  What  is  the  sum  of  30704  arid  471912  ? 

12.  What  is  the  sum  of  398463  and  401536  ? 

13.  If  a  top  costs  6  cents,  a  knife  25  cents,  a  slate   12 
cents  :  what  does  the  whole  amount  to  ? 

14.  John  gave  30  cents  for  a  bunch  of  quills,  18  cents  for 
an  inkstand,  25  cents  for  a  quire  of  paper :  what  did  the 
whole  cost  him  ? 

15.  If  2  cows  cost  143  dollars,  5  horses  621  dollars,  and  2 
yoke  of  oxen  124  dollars  :  what  will  be  the  cost  of  them  all  * 

16.  Add  5  units,  6  tens,  and  7  hundreds. 

ANALYSTS. — We  set  down  the  5  units  in  the  place       oi 
of  units,  the  6  tens  in  the  place  of  tens,  and  the  7 
hundreds  in  the  place  of  hundreds.  We  then  add  up,      "g  ^  JS 
and  find  the  sum  to  be  765. 

We  must  observe,  that  in  all  cases,  units  of  the  5 

same  order  are  written  in  the  same  column.  ^  6 

TT5" 


SIMPLE   NUMBERS.  25 

1  7.  What  is  the  sum  of  3  units,  8  tens,  and  4  thousands  ? 

18.  What  is  the  sum  of  8  hundreds,  4  tens,  6  units,  and  6 
thousands  ? 

19.  What  is  the  sum  of  3  units,  5  units,  6  tens,  3  tens,  4 
hundreds,  3  hundreds,  5  thousands,  and  4  thousands? 

20.  What  is  the  sum  of  five  units  of  the  4th  order,  1  of  the 
3d,  three  of  the  4th,  five  of  the  3d,  and  one  of  the  1st? 

21.  What  is  the  sum  of  six  units  of  the  2d  order,  five  of  the 
3d,  six  of  the  4th,  three  of  the  2d,  four  of  the  3d,  two  of  the 
1st,  and  four  of  the  2d? 

22.  What  is  the  sum  of  3  and  6,  5  tens  and  2  tens,  and  3 
hundreds  and  6  hundreds  ? 

23.  What  is  the  sum  of  4  and  5,  5  tens,  3  hundreds  and  2 
hundreds  ? 

GENERAL    METHOD. 

33.  1.  A  farmer  paid  898  dollars  for  one  piece  of  land,  and 
637  dollars  for  another;  how  many  dollars  did 
he  pay  for  both  ?  OPERATION. 

ANALYSIS.  —  Write  the  numbers  thus,  898 

and  draw  a  line  beneath  them. 


sum  of  the  units,        -  •  15 

sum  of  the  tens,          »  12 

sum  of  the  hundreds,  1  4 


sum  total  1535 

1.  The  example  may  be  done  in  another  way, 

thus :  Having  set  down  the  numbers,  as  before,  OPERATION. 

say,  7  units  and  8  units  are  15  units,  equal  to  898 

1  ten  and  5  units  :  set  the  5  in  the  units'  place,  63*7 

and  the  1  ten    in  the   column  of  tens.     Then  n 

say,  1  tea  and  3  tens  are  4  tens,  and  9  tens  are  1535 
13  tens,  equal  to  1   hundred  and  3  tens.     Set 

the  3  in  the   tens'  place  and  the  1  hundred  in  the  column  of 

33.  How  do  you  set  down  the  numbers  for  addition  ?  Where  do 
you  begin  to  add?  If  the  sum  of  any  column  can  be  expressed  by 
a  single  figure,  what  do  you  do  with  it?  When  it  cannot,  what  do 
you  write  down  ?  What  do  you  then  add  to  the  next  column  ?  When 
you  add  to  the  next  column,  what  is  it  called  ?  What  do  you  set 
down  when  you  come  to  the  last  column  ? 


26  ADDITION. 

hundreds.  Add  the  column  of  hundreds  and  write  down  the  sum, 
and  the  entire  sum  is  1535. 

~  2.  When  the  sum,  in  any  column,  exceeds  9,  it  produces  one  or 
more  units  of  a  higher  order,  which  belongs  to  the  next  column  at 
the  left.  In  that  case,  write  down  the  excess  over  exact  tens,  and 
add  to  the  next  column  as  many  units  of  its  own  order,  as  there 
were  tens  in  the  sum. 

This  is  called  carrying  to  the  next  column.  The  number  to 
be  carried,  should  not,  in  practice,  be  written  under  the  col- 
umn at  the  left,  but  added  mentally. 

Hence,  to  find  the  sum  of  two  or  more  numbers,  we  have 
the  following 

RULE. 

I.  Write  the  numbers  to  be  added,  so  that  units  of  the  same 
order  shall  stand  in  the  same  column. 

II.  Add  the  column  of  units.     Set  down  the  units  of  the 
sum  and  carry  the  tens  to  the  next  column. 

III.  Add  the  column  of  tens.     Set  down  the  tens  of  the  sum 
and  carry  the  hundreds  to  the  next  column ;  and  so  on,  till 
all  the  columns  are  added,  and  set  down  the  entire  sum  of  the 
last  column. 

PROOF. 

34*  The  proof  of  any  operation,  in  Addition,  consists  In 
showing  that  the  result  or  answer  contains  as  many  units  as 
there  are  in  all  the  numbers  added,  and  no  more.  There  are 
two  methods  of  proof,  for  beginners.* 

I.  Begin  at  the   top  of  the  units  column  and  add  all  the 
columns  downwards,  carrying  from  one  column  to  the  other, 
as   when    the   columns    were   added   upwards.      If  the    two 
results  agree   the  work  is  supposed   to  be  right.     For,  it  is 
not  likely  that  the  same  mistake  will  have  been  made  in  both 
additions. 

II.  Draw  a  line  under  the  upper  number.     Add  the  lower 
numbers  together,  and  then  add  their  sum  to  the  upper  number. 

*  NOTE. — If  the  teacher  prefers  the  method  of  proof  by  casting 
out  the  9's,  that  method,  for  the  four  ground  rules,  will  be  found 
in  the  University  Arithmetic. 

84.  What  does  the  proof  consist  of  in  addition?  How  many 
methods  of  proof  are  there?  Give  the  two  methods. 

NOTE. — Explain  the  process  of  addition  by  reading  the  figures. 


SIMPLE   NUMBERS. 


If  the  last  sum  is  the  same  as  the  svm  total,  first  found,  the 
work  may  be  regarded  as  right. 


EXAMPLES. 

1.  What  is  the  sum  of  the  numbers  375, 
6321,  and  598? 

The  small  figure  placed  under  the  4,  shows  how 
many  are  to  be  carried  from  the  units'  column,  and 
the  small  figure  under  the  9,  how  many  are  to  be 
carried  from  the  tens'  column. 

Also,  in  the  examples  below,  the  small  figure  un- 


OPERATION. 

375 

6321 

598 


7294 
11 

der  each  column  shows  how  many  are  to  be  carried  to  the  next 
column  at  the  left.  Beginners  should  set  down  the  numbers  to  be 
carried,  as  in  the  examples. 


Ans.  110012 

2221 


Ans. 


(3.) 

9841672 
793159 

888923 

11523754 

221111 


(4.) 
81325 
6784 
2130 

Ans.  90239 
1110 


(5.) 
4096 
3271 
4722 


(6.) 
9976 

8757 
8168 


9875 
9988 

8774 


(8.) 
67954 
98765 
37214 


(9.) 
6412 
1091 
6741 

9028 


(10.) 
90467 
10418 
91467 
41290 


(11.) 
87032 
64108 
74981 
21360 


(12.) 
432046 
210491 

809765 
542137 


(13.) 

21467 

80491 

67421 

4304 

2191 


(14.) 

89479 

75416 

7647 

214 

19 


(15.) 

74167 

21094 

2947 

674 

85 


(16.) 

9947621 

704126 

81267 

9241 

495 


28 


ADDITION. 


(17.) 

34578 

~3750 

87 

328 

17 

327 

Sum  39087 
~4509 


Proof  39087 

(20.) 

672981043 

67126459 

39412767 

7891234 

109126 

84172 

72120 


(18.) 

22345 

67890 

8752 

340 

350 

78 


Sum  99755 


77410 
Proof  1)9755 

(21.) 

91278976 

7654301 

876120 

723456 

31309 

4871 

978 


(19.) 

23456 

78901 

23456 

78901 

23456 

78901 

Sum  307071 


Proof  307071 

(22.) 

8416785413 

6915123460 

31810213 

7367985 

654321 

37853 

2685 


READING. 

The  pupil  should  be  early  taught  to  omit  the  intermediate  wordi 
in  the  addition  of  columns  of  figures.  Thus,  in  example  22, 
instead  of  saying  5  and  8  are  eight  and  1  are  nine,  he  should  say 
eight,  nine,  fourteen,  seventeen,  twenty.  Then,  in  the  column  of 
tens,  ten,  fifteen,  seventeen,  twenty-five,  twenty-six,  thirty-two, 
thirty-three.  This  is  called  reading  the  columns.  Let  the 
pupils  be  often  practised  in  it,  both  separately,  and  in  concert  in 
classes. 

23.  Add   8635,   2194,   7421,   5063,  2196,  and    1245    to- 
gether. 

24.  Add   246034,   29S765,  47321,   58653,   64218,    5376, 
9821,  and  340  together. 

25.  Add   27104,   32547,    10758,   6256,   704321,   730491, 
2587316,  and  2749104  together. 

26.  Add   1,  37,  39504,  6890312,  18757421,  and   265  to- 
gether. 

27.  What   is   the    sum    of   the    following   numbers,    via: 
seventy-five;    one   thousand   and   ninety-five;    six   thousand 
four  hundred  and  thirty-five;  two  hundred  and  sixty-seven 


SIMPLE   NUMBERS.  29 

thousand  ;  one  thousand  four  hundred  and  fifty-five  ;  twenty- 
seven  millions  and  eighteen  ;  two  hundred  and  seventy  mil- 
lions and  twenty-seven  thousand  ? 

28.  What    is    the    sum    of    372856,    404932,    2704793, 
9078961,  304165,  207708,  41274,  375,  271,  34,  and  6? 

29.  What  is   the   sum  of  4073678,  4084162,  3714567, 
27413121,  27049,  87419,  27413,  604,  37,  and  9  ? 

30.  What  is  the  sum  of  36704321,  2947603,  999987,  76, 
47213694,  21612090,  8746,  31210496,  and  3021  ? 

31.  Add  together    fifty-eight  billions,  nine  hundred  and 
eighty-two  mill  ions,  four  hundred  and  eighty-seven  thousands, 
six  hundred  and  fifty-four  ,-  seven  hundred  and  forty  billions, 
three  hundred   and   fifty   millions,   five   hundred  and   forty 
thousands,   seven   hundred   and    sixty ;    four   hundred   and 
twenty-five  billions,  seven  hundred  and  three  millions,  four 
hundred  and  two  thousands,  six  hundred  and  three ;  thirty- 
four  billions,  twenty  millions,   forty  thousands  and  twenty  ; 
five  hundred  and  sixty  billions,  eight  hundred  millions,  seven 
hundred  thousands  and  five  hundred. 

(32.)  (33.)  (34.) 

87406  92674  25043 

89507  27049  97069 

41299  28372  81216 

47208  37041  75850 

71615  49741  90417 

72428  57214  19216 

97206  59261  20428 

41278  41219  60594 

28907  57267  72859 

325412  3  40216  §  43706 

S 27049  g 87614  g 21441 

28416  92742  87604 

72204  87046  71215 

70412  90212  .  18972 

27426  17618  27042 

62081  40261  59876 

81697  57274  54301 

87489  21859  87415 

21642  42673  32018 

24672  51814  7268T 


30  ADDITION. 

APPLICATIONS. 

35*  In  all  the  applications  of  arithmetic,  the  numbers  ad- 
ded together  must  Imve  the  same  unit. 

In  the  question,  How  many  head  of  live  stock  in  a  field, 
there  being  6  cows,  2  oxen,  3  steers,  and  15  sheep,  the  unit 
is  1  head  of  live  stock.  And  the  same  principle  is  applicable 
to  all  similar  questions. 

QUESTIONS    FOR    PRACTICE. 

1.  HOTT  many   days   are   there   in   the    twelve   calendar 
months?     January  has  31,  February  28,  March   31,  April 
30,  May  31,  June  30,  July  31,  August  31,  September  30, 
October  31,  November  30,  and  December  31. 

Ans. 

2.  What  is  the  total  weight  of  seven  casks  of  merchandise ; 
No.  1,  weighing   960   pounds,  No.  2,  725  pounds,  No.  3, 
830  pounds,  No.  4,  798  pounds,  No.  5,  698  pounds,  No.  6, 
569  pounds,  No.  7,  987  pounds  ? 

3.  At  the  Custom   House,  on  the  1st  day  of  June,  there 
ir ere  entered  1800  yards  of  linen;  on  the  10th,  2500  yards; 
on  the  25th,  600  yards;  on  the  day  following,  7500  yards; 
and  the  last  three  days  of  the  month,  1325  yards  each  day : 
•what  was  the  whole  amount  entered  during  the  month  ? 

Ans. 

4.  A  farmer  has  his  live-stock  distributed  in  the  following 
manner:  in  pasture  No.  1,  there  are  5   horses,  14  cows,  8 
oxen,  and  6  colts ;  in  pasture  No.  2,  3  horses,  4  colts,  6  cows, 
20  calves,  and  12  head  of  young  cattle;  in  pasture  No.  3, 
320  sheep,  16  calves,  two  colts,  and  5  head  of  young  cattle. 
How  much  live-stock  had  he  of  each  kind,  and  how  many 
Lead  had  he  altogether  ? 

Ans.       horses,       cows,       oxen,       colts,       calves, 
head  of  young  cattle,  and  sheep. 

Total  live-stock,  head. 

5.  What  is  the  interval  of  time  between  an  event  which 
happened  125  years  ago,  and  one  that  will  happen  267  years 
hence  ? 

6.  There  are  60  seconds  in  a  minute,  3600  in  an  hour, 

35.  What  principles  govern  all  the  additions  in  Arithmetic  ?  What 
is  the  unit  in  the  question  ?    How  many  head  of  cattle  in  a  pasture  ? 


SIMPLE  NUMBERS.  81 

86400  in  a  day,  604800  in  a  week,  2419200  in  a  month, 
and  31557600  in  a  year:  how  many  seconds  in  the  time 
named  above  ? 

7.  Suppose  •  a  merchant  to  buy  the  following  parcels  of 
cloth:    3912*  yards,   1856,   2011,   4540,   937,   6338,   3603, 
1586,2044,2951,4228,    1345,    1011,6138,960,607,5150,*, 
13886,   617,  7513,   4079,  743,   612,  2519,  1238,   and  2445 
yards :  how  many  yards  in  all  ? 

8  What  is  the  sum  of  two  millions  bushels  of  corn,  five 
hundred  and  thirty-one  thousand  bushels,  one  hundred  and 
twenty  bushels,   fourteen  thousand  bushels,  thirty  thousand 
and  twenty  four  bushels,  five  hundred  and  sixty  bushels,  and 
seven  hundred  and  two  bushels  ? 

9  The  mail  route  from  Albany  to  New  York  is  144  miles, 
from  New  York  to  Philadelphia  90  miles,  from  Philadelphia 
to  Baltimore  98  miles,  and  from  Baltimore  to  Washington 
City  38  miles :  what  is  the  distance  from  Albany  to  Washing- 
ton'? 

10.  A  man  dying  leaves  to  his  only  daughter  nine  hundred 
and  ninety-nine  dollars,  and  to  each  of  three  sons  two  hundred 
dollars  more  than  he  left  the  daughter.     What  was  each  son's 
portion,  and  what  the  amount  of  the  whole  estate  ? 

A        ( Each  son's  part         dollars. 
''  \  Whole  estate  dollars. 

11.  The  number  of  acres  of  the  public  lands  sold  in  1834 
was  4658218  ;  in  1835,  12564478  ;  in  1836,  25167833     The 
number  sold  in    1840  was  2236889;  in   1841,  1164796;  in 
1842,  1 129217      How  many  acres  were  sold  in  the  first  three, 
and  how  many  in  the  last  three  years  ? 

A        C  1st  3  yrs. 
Ans  \  last      " 

12    What  was  the  population  of  the  British  provinces  in 
North  America  in  1834,  the  population  of  Lower  Canada 
being  stated  at  549005,  of  Upper  Canada  336461,  of  New  ,< 
Brunswick  152156,  of  Nova  Scotia  and  Cape  Breton  142548, ' 
of  Prince  Edward's  Island  32292,  of  Newfoundland  75000  ? 

Ans. 

13.  By  the  census  of  1850,  the  population  of  the  ten 
largest  cities  was  as  follows  :  New  York  515547  ;  Philadelphia 
340045  ;  Baltimore  169054  ;  Boston  136881  ;  New  Orleans 
116375;  Cincinnati  115436;  Brooklyn  96838;  St.  Louis 


32  ADDITION. 

77860;  Albany  50763;  Pittsburgh  46601:  what  was  their 
entire  population  ? 

14.  By  the  census  of  1850,  the  number  of  deaf  and  dumb 
in  the  United  States  was  9803  ;  of  blind  9794  ;  of  insane 
15610  ;  of  idiots  15787  :  what  was  the  aggregate  ? 


15.  By  the  census  of  1850,  the  population  of  the  District 
of  Columbia  was  51687  ;  of  the  Territory  of  Minnesota 
6077  ;  of  New  Mexico  61547  ;  of  Oregon  13294  ;  of  Utah 
11380  :  what  was  the  population  of  the  Territories,  including 
the  District  of  Columbia  ? 

16  By  the  census  of  1850,  the  population  of  Maine  was 
583169;  of  New  Hampshire  3L7976;  of  Vermont  314120; 
of  Massachusetts  994514  ;  of  Rhode  Island  147545  ;  and  of 
Connecticut  370792:  what  was  the  population  of  the  six 
New  England  States  ? 

17.  By  the  census  of  1850,  the  population  of  New  York 
was  3097394  ;  the  population  of  New  Jersey  489555  ;  oi 
Pennsylvania  2311786;  and  of  Delaware  91532  :  what  was 
the  population  of  the  four  Middle  States  ? 

18.  By  the  census  of  1  850,  the  population  of  Maryland  was 
583034  ;  of  Virginia  1421661  ;  of  North  Carolina  869039  ; 
of  South  Carolina  668507  ;  of  Georgia  906185;  of  Florida 
87445;    of  Alabama   771623;    of  Mississippi  606526;    of 
Louisiana  517762;    and  of  Texas  212592:  what  was  the 
whole  population  of  the  ten  Southern  States  ? 

Ans. 

19.  By  the  census  of  1850,  the  population  of  Tennessee 
was  1002717;  of  Kentucky  982405;  of  Ohio  1980329;  of 
Indiana  988416;  of  Illinois  851470;  of  Michigan  397654; 
of  Wisconsin  305391  ;  of  Iowa  192214  ;  of  Missouri  682044  ; 
of  Arkansas  209897  ;  and  of  California  92597  :  what  was  the 
entire  population  of  the  eleven  Western  States  ? 

Ans* 

20.  By  the  census  of  1850,  the  population  of  the  six  New 
England  States  was  2728116;  of  the  four  Middle  States 
5990267  ;  of  the  ten  Southern  States  6644374  ;  of  the  eleven 
Western  States  7685134  ;  and  of  the  five  Territories  143985  : 
what  was  the  entire  population  ? 

21.  Write  the  population  of  each  State  and  Territory,  in 
eluding  the  District  of  Columbia,  and  add  the  whole  as  ft 
single  example. 


SUBTRACTION. 


SUBTRACTION. 

86*  1.  John  has  3  apples  and  Charles  has  2 :  how  many 
have  both  ? 

If  John's  apples  be  taken  from  the  sum,  5  apples,  how 
many  apples  will  remain  ?  2  from  5  leaves  how  many  f 

2.  If  James  has  5  apples  and  gives  3  to  Charles,  how 
many  will  he  have  left  ?  3  from  5  leaves  how  many  $ 

Let  the  following  table  be  carefully  committed  to  memory: 

SUBTRACTION  TABLE. 


1  from  1  leaves  0 
1  from  2  leaves  1 
1  from  3  leaves  2 
1  from  4  leaves  3 
1  from  5  leaves  4 
1  from  6  leaves  5 
1  from  7  leaves  6 
1  from  8  leaves  7 
1  from  9  leaves  8 
1  from  10  leaves  9 
1  from  11  leaves  10 

2  from  2  leaves  *0 
2  from  3  leaves  1 
2  from  4  leaves  2 
2  from  5  leaves  3 
2  from  C  leaves  4 
2  from  7  leaves  5 
2  from  8  leaves  6 
2  from  9  leaves  7 
2  from  10  leaves  8 
2  from  11  leaves  9 
2  from  12  leaves  10 

3  from  3  leaves  0 
3  from  4  leaves  1 
3  from  5  leaves  2 
3  from  6  leaves  3 
3  from  7  leaves  4 
3  from  8  leaves  5 
3  from  9  leaves  6 
3  from  10  leaves  7 
3  from  11  leaves  8 
3  from  12  leaves  9 
3  from  13  leaves  10 

4  from  4  leaves  0 
4  from  5  leaves  1 
4  from  6  leaves  2 
4  from  7  leaves  3 
4  from  8  leaves  4 
4  from  9  leaves  5 
4  from  10  leaves  6 
4  from  11  leaves  7 
4  from  12  leaves  8 
4  from  13  leaves  9 
4  from  14  leaves  10 

5  from  5  leaves  0 
5  from  C  leaves  1 
5  from  7  leaves  2 
5  from  8  leaves  3 
5  from  9  leaves  4 
5  from  10  leaves  5 
5  from  11  leaves  0 
5  from  12  leaves  7 
5  from  13  leaves  8 
5  from  14  leaves  9 
5  from  15  leaves  10 

6  from  6  leaves  0 
6  from  7  leaves  1 
6  from  8  leaves  2 
6  from  9  leaves  3 
6  from  10  leaves  4 
6  from  11  leaves  5 
6  from  12  leaves  6 
0  from  13  leaves  7 
6  from  14  leaves  8 
6  from  15  leaves  9 
C  from  16  leaves  10 

7  from  7  leaves  0 
7  from  8  leaves  1 
7  from  9  leaves  2 
7  from  10  leaves  3 
7  from  11  leaves  4 
7  from  12  leaves  5 
7  from  13  leaves  6 
7  from  14  leaves  7 
7  from  15  leaves  8 
7  from  16  leaves  9 
7  from  17  leaves  10 

8  from  8  leaves  0 
8  from  9  leaves  1 
8  from  10  leaves  2 
8  from  11  leaves  3 
8  from  12  leaves  4 
8  from  13  leaves  5 
8  from  14  leaves  6 
8  from  15  leaves  7 
8  from  16  leaves  8 
8  from  17  leaves  9 
8  from  18  leaves  10 

9  from  9  leaves  0 
9  from  10  leaves  1 
9  from  11  leaves  2 
9  from  12  leaves  3 
9  from  13  leaves  4 
9  from  14  leaves  5 
9  from  15  leaves  6 
9  from  16  leaves  7 
9  from  17  leaves  8 
9  from  18  leaves  9 
9  from  19  leaves  10 

34  SUBTRACTION. 

PRINCIPLES    AND    EXAMPLES. 

37 »  John  has  6  apples  and  gives  4  to  Charles :  how  many 
has  he  left  ? 

The  2  is  called  the  difference  between  the  numbers  6 
and  4  •  and  this  difference  added  to  the  less  number  4,  will 
give  the  greater  number  6  :  hence, 

"  THE  DIFFERENCE  between  two  numbers,  is  such  a  number  as 
(added  to  the  less  will  give  the  greater. 

SUBTRACTION  is  the  operation  of  finding  the  difference  be- 
tween two  numbers. 

When  the  two  numbers  are  unequal,  the  larger  is  called 
the  minuend,  and  the  less  is  called  the  subtrahend.  Their 
difference,  whether  they  are  equal  or  unequal,  is  called  the 
remainder.  — 

OF   THE    SIGNS. 

38»  The  sign  — ,  is  called  minus,  a  term  signifying  less. 
When  placed  between  two  numbers  it  denotes  that  the  one 
on  the  right  is  to  be  taken  from  the  one  on  the  left. 

Thus,  6—4=2,  denotes  that  4  is  to  be  taken  'from  6.  Here, 
6  is  the  minuend,  4  the  subtrahend,  and  2  the  remainder. 


12—2  = 


—   = 

12  —  3=  how  many  ? 
16  —  4=  how  many  ? 
11  —  6=  how  many  ? 
18  —  9=  how  many? 
25  —  8=  how  many  ? 


17  —  7=  how  many? 
16  —  8=  how  many  ? 
19  —  9=  how  many? 
20 — 4=  how  many  ? 
13—7=  how  many? 
14  —  2=  how  many? 


EXAMPLES. 

1.  James  has  27  apples,  and  gives  14  to  John :  how  many 
has  he  left? 

37.  What  is  the  difference  between  two  numbers  ?    What  is  Sub- 
traction ?     What  is  the  larger  number  called  ?    What  is  the  smaller 
number  called  ?     What  is  the  difference  called  ?    In  the  first  exam- 
ple, which  number  was  the  minuend  ?    Which  the  subtrahend  ? 
Which  the  remainder? 

38.  What  is  the  sign  of  Subtraction  ?    What  is  it  called  ?    What 
does  the  term  signify  ?    When  placed  between  two  numbers  what 
does  it  denote  ? 


SIMPLE  NUMBERS.  35 

The  27  is  made  up  of  7  units  and  2  tens;  27  Minuend, 

and  the  14,  of  4  units  and  1  ten.     Subtract  4  **    Q  ,.    ,       -. 

unite  from  7  units,  and  3  units  will  remain;  ±2 

subtract  1  ten  from  2  tens  and  1  ten  will  re-  13  Bemamder. 
main :  hence,  the  remainder  is  13. 

2.  What  are  the  remainders  in  the  following  examples  : 

(1.)  (2.)  (3.  (4.) 

Minuends,              874  972  999  8497 

Subtrahends,             642  '    631  367  7487 

Remainders,  232  1010 

3.  A  farmer  had  378  sheep,  and  sold  256  :  how  many  had 
he  left? 

We  first  write  the  number  378,  and  then  256  under  373 

it,  so  that  units  of  the  same  order  shall  fall  in  the  same  2  z.a 

column.    We  then  take  6  units  from  the  8  units,  5  tens  __ 

from  7  tens,  and  2  hundreds  from  3  hundreds,  leaving  for  122 
the  remainder  122. 

4.  A  merchant  had  578  dollars  in  cash,  and  paid  475  dol- 
lars for  goods :  now  much  had  he  left  ? 

5.  What  are  the  remainders  in  the  following  examples : 

(1.)  (2.)                                 (3.) 

62843  278846  894862 

51720  167504  170641 
Tll23 


39,  We  see,  from  the  above  examples, 

1st.  That  units  of  the  same  order  are  written  in  the 
same  column  ;  and 

2d.  That  units  of  any  order  are  always  subtracted  from 
units  of  the  same  order. 

40.  To  find  the  difference  when  any  figure  of  the  minuend 
is  less  than  the  one  which  stands  under  it. 

1.  What  is  the  difference  between  843  and  562  ? 

39.  What  principles  are  shown  by  the  examples  ? 

40.  Can  you  subtract  a  greater  number  from  a  less?    When  the  tipper  figure 
is  the  least,  how  do  you  proceed?    Does  this  change  the  difference  between  the 
numbers  ?    What  then  may  we  always  do  ? 


36  SUBTK  ACTION. 

ANALYSIS. — Begin  at  the  units'  column,  and  say,  OPERATION. 
2  from  3  leaves  1,  which  is  written  in  the  units'          g^o 
place.     At  the  next  place  we  meet  a  difficulty,  for 
we  cannot  subtract  a  greater  number  from  a  less. 

If  now,  we  take  1  from  the  8  hundreds  (equal  to 
f  10  tens)  and  add  it  to  the  4  tens,  the  minuend  will 
become  7  hundreds,  14  tens,  and  3  units,  as  written 
below.  We  may  then  say  6  tens  from  14  tens  leaves 
8  tens ;  and  then  5  hundreds  from  7  hundreds  leaves 
2  hundreds ;  hence,  the  remainder  is  281. 

The  same  result  is  obtained  by  adding,  mentally,  10  to  1  o 
the  4  tens,  and  then  adding  1  to  5,  the  next  figure  of  the 

subtrahend  at  the  left ;  for,  adding  1  to  the  5  is  the  same  562 

as  diminishing  the  8  by  1.     This  process  of  adding  10     _J 

to  a  figure  of  the  minuend  and  returning  1  to  the  next  281 
figure  of  the  subtrahend,  at  the  left,  is  called  'borrowing. 

41*  Hence,  to  find  the  difference  between  two  numbers,  we 
have  the  following 

KULE. 

I.  Set  down  the  less  number  under  the  greater,  so  that  units 
of  the  same  order  shall  fall  in  the  same  column. 

IL  Begin  at  the  right  hand  •  subtract  each  figure  of  the 
lower  line  from  the  one  directly  over  it,  when  the  upper 
figure  is  the  greater;  but  when  it  is  the  less,  add  10  to  it, 
before  subtracting,  after  which  add  1  to  the  next  figure  of  the 
subtrahend. 

PROOF. 

The  remainder  or  difference  is  such  a  number  as  added  to 
the  subtrahend,  will  give  a  sum  equal  to  the  minuend,  (Art. 
§7,)  hence : 

Add  the  remainder  to  the  subtrahend.  If  the  work  is  right 
$IK  sum  will  be  equal  to  the  minuend. 

EXAMPLES.          d^ 


Minuends, 
Subtrahends, 
Remainders, 
Proofs, 

(1.) 

8592678 

1078953 

J2-> 

67942139 

9756783 

(3.) 
219067803 
104202196 

7513725 

8592678 

67942139 

219067803 

41.  How  do  you  set  down  the  numbers  for  subtraction  ?  Where 
do  you  begin  to  subtract  ?  How  do  you  subtract  ?  Give  the  rule  ? 
How  do  you  prove  subtraction? 


SIMPLE  NUMBERS.  37 

(4.)    (5.)    (6.)  (7.)    (8.) 

10000  30000  67087  100000  87000 

4   9999  40000  1   1009 

Remainders,      9996  85991 


9.  From  2637804  take  2376982. 

10.  From  3762162  take  826541. 

11.  From  78213609  take. 27821890. 

12.  From    thirty    thousand    and    ninety-seven,   take   one 
thousand  six  hundred  and  fifty-four. 

13.  From  one  hundred  millions  two  hundred  and  forty-seven 
thousand,  take  one  million  four  hundred  and  nine. 

14.  Subtract  one  from  one  million. 

15.  From  804367  subtract  27905. 

16.  From  18623041  subtract  61294. 

17.  From  4270492  subtract  26409. 

18.  From  8741209  subtract  728104. 

19.  From  741874  subtract  689346. 

SPELLING READING. 

42.  1.  What  is  the  difference  between  725  and  341  ? 

OPERATION. 

By  the  common  method,  which  is  spelling,  we  say,  725 
1  from  5  leaves  4 ;  4  from  12  leaves  8  ;  1  to  carry  34^ 
to  3  is  4 ;  4  from  7  leaves  3. 

Reading  the  words  which  express  the  final  result,  we  should 
make  the  operations  mentally,  and  say,  4,  8,  3. 

Let  the  pupils  be  practiced  separately  in  the  reading,  and  also 
in  concert  in  classes. 

APPLICATIONS. 

43.  It  should  be  observed,  that  in  all  the  applications  of 
Subtraction,  one  number  can  be  subtracted  from  another,  only 
when  they  both  have  the  same  unit. 

. 

42.  Explain  the  process  of  reading  the  results  in  subtraction. 

43.  What  is  always  necessary  in  order  that  one  number  may  be 
subtracted  from  another  ? 


38  SUBTEACTIOK. 

EXAMPLES    FOR  PRACTICE. 

1.  Suppose  John    were    Lorn    in    eighteen    hundred    and 
fifteen,  and  James   in    eighteen    hundred   and   twenty-five : 
what  is  the  difference  of  their  ages  ? 

2.  A  man  was  born  in  1785 :  what  was  his  age  in  1830  ? 

Ans. 

3.  Suppose  I  lend  a  man  1565  dollars,  and  he  dies,  owing 
me  450  dollars  :  how  much  had  he  paid  me? 

Ans, 

4.  In  five  bags  are  different  sums  of  money  to  the  amount 
in  all  of  1000  dollars.     In  the  first  there  are  100  dollars;  in 
the  second,  314  dollars;  in  the  third,  143  dollars ;  and  in  the 
fourth,  209  dollars  :  how  many  dollars  does  the  fifth  contain  ? 

Ans. 

5.  America  was  discovered  by  Christopher    Columbus  in 
the  year  1492.     What  number  of  years  has  since  elapsed  ? 

6.  George    Washington  was  born  in  the  year   1732,   and 
died  in  1799  :  how  old  was  he  at  the  time  of  his  death  ? 

Ans. 

7.  The  declaration  of  independence  was   published,  July 
4th,  1776:  how  many  years  to  July  4th,  1838? 

Ans. 

8.  In  1850  there  were  in  the  State  of  New  York  3,097,394 
inhabitants,  and  in  the  State  of  Pennsylvania  2,311,786  in- 
habitants: how  many  more  inhabitants  were  there  in  New 
York  than  in  Pennsylvania  ?  Ans. 

9.  The  revolutionary  war  began  in  1775  ;  the  next  war  in 
1812  :  what  time  elapsed  between  their  commencements? 

Ans. 

10.  In  1850  there  were  in  New  York,  which  is  the  largest 
city  in  the  United  States,  515,547  inhabitants,  and  in  Phila- 
delphia, the  next   largest  city,    340,045:    how  many    more 
inhabitants  were  there  in  New  York  than  in  Philadelphia  ? 

Ans. 

11.  A  man  dies  worth  1200  dollars:  he  leaves  504  to  his 
daughter,  and  the  remainder  to  his  son?    what  was  the  son's 
portion  ? 

12.  Suppose  a  gentleman  has  an  income  of  3090  dollars 
a  year,  and   pays  for  taxes  150  dollars,  and  expends  besides 
307  dollars:  how  much  does  he  save? 


SIMPLE  NUMBERS.  39 

IS.  A  merchant  bought  500  barrels  of  flour  for  3500  dol- 
lars; he  sold  250  barrels  for  2000  dollars:  how  many  bar- 
rels remained  on  hand,  and  how  much  must  he  sell  them  for, 
that  he  may  lose  nothing  ? 

14.  The  tune  of  Yankee  Doodle  was  composed  by  a  doctor 
of  the  British  Army  to  ridicule  the  Americans  in  1775  :  how 
many  years  to  the  present  time  ? 

15.  Lord  Corn wallis  surrendered  at  Yorktown,  and  marched 
into  the  American  lines  in  1781  to  the  tune  of  Yankee  Doodle: 
how  many  years  was  that  after  the  tune  was  composed? 

Am. 

16.  At  a  certain  period  there  were  4338472  children  in 
the  United  States  between  the  ages  of  5  and  ]5;   of  this 
number  2477667  were  in  schools:   how  many  were  out  of 
schools? 

17.  The  circulation  of  the  blood  was  discovered  in  1616: 
how  many  years  to  1855? 

18.  Henry  Hudson  sailed  up  the  Hudson  river  in  1609: 
how  many  yean,  since? 

19.  Pliny  the  historian  died  17  years  after  the  birth  of 
Christ:   how  many  years  before  the  declaration  of  independ- 
ence ?  Ans. 

20.  Potatoes  were  carried  to  Ireland  from  America  in  1565 : 
how  many  years  was  that  before  the  settlement  of  Plymouth 
in  1620? 

21.  The  Mariner's  Compass  was  discovered  in  England  in 
the  year  1302  :  how  many  years  was  this  before  the  discovery 
of  America  in  1492  ?     How  many  years  to  the  present  time? 

Ans. 

22.  A  merchant  bought  1675  yards  of  cloth,  for  which  he 
paid  5025  dollars:  he  then  sold  335  yards  for  1005  dollars; 
how  much  had  he  left,  and  what  did  it  cost  him  ? 

Ans. 

23.  In  1850  the  slaves  in  the  United  States  amounted  to 
3204313;   free  colored  to  434495:    what  was  their  differ- 
ence? 

24.  What  length  of  time  elapsed  between   the  birth   of 
William  Penn  in  1644  and  the  birth  of  Sir  William  Herschel 
in  1738? 


40  SUBTRACTION. 

25.  What  length  of  time  elapsed  between  the  birth  of  Sir 
Francis  Bacon  in  1561  and  the  birth  of  Benjamin  Franklin 
in  1706? 

26.  What  length  of  time   elapsed   between   the  birth   of 
Shakespeare  in  1564  and  the  birth  of  George  Washington  in 
1732? 

27.  What  length  of  time  elapsed  between  the  birth  of  John 
Milton  in  1608  and  the  Declaration  of  Independence  in  1776? 

28.  What   length  of  time   elapsed   between   the  birth  of 
Oliver  Cromwell  in  1599  and  the  birth  of  Patrick  Henry  in 
1736? 

29.  By  the  census  of  1850,  the  number  of  white  inhabitants 
in  the  United  States  amounted  to  19553068 ;  and  the  blacks 
to  3638808 :  by  how  many  did  the  white  inhabitants  exceed 
the  black  ? 

30.  By  the   census  of  1850,  the  entire  population  of  the 
United  States  was  23191876;  that  of  the  six  New  England 
States,  2728116:   by  how   many   did  the  whole  population 
exceed  that  of  the  six  New  England  States  ? 

31.  In  1850,  the  slaves  in  the  United  States  amounted  to 
3204313;   and  the  free   colored  to  434495:   what  was  their 
difference  ? 

APPLICATIONS    IN    ADDITION    AND    SUBTRACTION. 

1.  A  merchant  buys  19576   yards  of  cloth  of  one  person, 
27580  yards  of  another,  and  375  of  a  third  ;   he  sells  1050 
yards  to  one  customer,  6974   yards  to  another,  and  10462 
yards  to  a  third :  how  many  yards  has  he  remaining  ? 

Ans. 

2.  A  person  borrowed  of  his   neighbor  at  one  time   355 
dollars,  at  another  time  637  dollars,  and  403  dollars  at  another 
time;  he  then  paid  him  977  dollars;  how  much  did  he  owe 
him? 

3.  I  have  a  fortune  of  2543  dollars  to  divide  amoncj  my 
four  sons,  James,  John,   Henry  and   Charles.     I  give  James 
504  dollars,  John    600    dollars,  and  Henry  725 :  how  much 
remains  for  Charles? 

4.  I  have  a  yearly  income  of  ten  thousand  dollars.     I  pay 
275  for  rent,  220  dollars  for  fuel,  35  dollars  to  the  doctor,  and 
3675  dollars    for  all   my  other  expenses:   how  much  have  I 
left  at  the  end  of  the  tyear  ? 


SIMPLE  NUMBERS.  41 

5.  A  man  pays  300  dollars  for  100  sheep,  95  dollars  for  a 
pair  of  oxen,  60  dollars  for  a  horse,  and  125  dollars  for  a  chaise. 
He  gives  100  bushels  of  wheat  worth  125  dollars,  a  cow  worth 
25  dollars,  a  colt  worth  40  dollars,  and  pays  the  rest  in  cash : 
how  much  money  does  he  pay  ? 

6.  A  merchant  owes  450120  dollars,  and  has  property  as 
follows :  bank  stock  350000  dollars,  western  lands  valued  at 
225100,  furniture  worth  4000   dollars,  and  a  store  of  goods 
worth  96000:  how  much  is  he  worth? 

Ans. 

7.  If  a  man's  income  is  3467  dollars  a  year,  and  he  spends 
269  dollars  for  clothing,  467  for  house  rent,  879  for  provi- 
sion, and  146  for  travelling:  how  much  will  he  have  left  at 
the  end  of  the  year? 

8.  A  man  gains  367   dollars,  then   loses  423  ;   a  second 
time  he  gains  875  and  loses  912  ;  he  then  gains  1012  dollars  ; 
how  much  more  has  he  gained  than  lost? 

9.  If  I  agree  to  pay  a  man  36  dollars  for  plowing  25  acres 
of  land,  200  dollars  for  fencing  it,  and  150  for  cultivating  it, 
how  much  shall  I  owe  him  after  paying  331  dollars  ? 

Ans. 

10.  A  merchant  bought  85  hogsheads  of  sugar  for  28675 
dollars,  paid  1231  dollars  freight,  and  then  sold  it  for  1683 
dollars  less  than  it  cost  him  :  how  much  did  he  receive  for  it? 

11.  If  I  buy  489  oranges  for  912  cents,  and  sell  125  for 
186.  cents,  and  then  sell   134  for  199   cents,  how  many  will 
be  left,  and  how  much  will  they  have  cost  me  ? 

12.  By  the  census  of  1850,  the  entire  population  of  the 
United  States  was  23191876  ;  the  slave  population  3204313  ; 
free  colored  434495  :  what  was  the  white  population  ? 

Ans. 

13.  Six  men  bought  a  tract  of  land  for  36420  dollars:  the 
first  man  paid  12140  ;  the  second  3035  less  than  the  first;  the 
third  346  ;  the  fourth  6070  more  than  the  third  ;  the  fifth  1821 
less  than  the  fourth  :  how  much  did  the  sixth  man  pay  ? 

14.  The   coinage    in    the  United    States   Mint    from    its 
establishment  in    the  year  1792  to   1836   was    thus:    gold 
22102035  dollars;  silver  46739182  dollars;  copper  740331 
dollars.     The  amount  coined  from    the  year  1837  to  1848 
was  81436165  dollars:  how  much   more'was  coined  in  the 
last  mentioned  period  than  in  the  first? 


MULTIPLICATION. 


MULTIPLICATION. 

44.  1.  If  Charles  gives  2  cents  apiece  for  two  oranges,  how 
much  do  they  cost  him  ? 

2.  If  Charles  gives  2  cents  apiece  for  three   oranges,  how 
much  do  they  cost  him  ? 

3.  If  he  gives  2  cents  apiece  for  4  oranges,  how  much  do 
they  cost  him  ? 

The   cost,  in    each  case,  may  be  obtained  by  adding  the 
price  of  a  single  orange : 

.2  +  2  =  4  cents,  the  cost  of  2  oranges. 
2+2+2=6   cents,  the  cost  of  3  oranges. 
2  +  2  +  2  +  2  =  8  cents,  the  cost  of  4  oranges. 
In  toe  first  case  2  is  taken  two  times  ;  in  the  second,  three 
times;  in  the  third,  four  times;  and  any  number  may  be 
repeated  by  adding  it  continually  to  itself. 

MULTIPLICATION  TABLE. 


Once           0  is    0 

3  times     0  are     0 

5  times     0  are     0 

Once           1  is     1 

3  times     1  are     3 

5  times     1  are     5 

Once            2  is     2 

3  times     2  are     6 

5  times     2  are  10 

Once            3  is     3 

3  times     3  are     9 

5  times     3  are  15 

Once           4  is    4 

3  times     4  are  12 

5  times    4  are  20 

Once            5  is     5 

3  times     5  are  15 

5  times     5  are  25 

Once            6  is     G 

3  times     6  are  18 

5  times     6  are  30 

Once            7  is     7 

3  times     7  are  21 

5  times     7  are  35 

Once            8  is     8 

3  times     8  are  24 

6  times     8  are  40. 

Once            9  is     9 

3  times     9  are  27 

5  times     9  are  45 

Once          10  is  10 

3  times  10  are  30 

5  times  10  are  50 

Once          11  is  11 

3  times  11  are  33 

5  times  1  1  are  55 

Once          12  is  12 

3  times  12  are  36 

5  times  12  are  60 

2  times     0  are     0 

4  times     0  are     0 

6  times     0  are     0 

2  times     1  are     2 

4  times     1  are     4 

6  times     1  are     6 

2  times     2  are    4 

4  times     2  are     8 

6  times     2  are  12 

2  times     3  are     6 

4  times     3  are  12 

6  times     3  are  18 

2  times     4  are     8 

4  times     4  are  16 

6  times     4  are  24 

2  times     5  are  10 

4  times     5  are  20 

6  times     5  are  30 

2  times     6  are  12 

4  times     6  are  24 

6  times     6  are  36 

2  times     7  are  14 

4  times     7  are  28 

6  times     7  are  42 

2  times     8  are  16 

4  times     8  are  32 

6  times     8  are  48 

2  times    9  are  18 

4  times     9  are  36 

6  times     9  are  54 

2  times  10  are  20 

4  times  10  are  40 

6  times  10  are  60 

2  times  11  are  22 

4  times  11  are  44 

6  times  11  are  66 

2  times  12  are  24 

4  times  12  are  48 

6  times  12  are  72 

SIMPLE  NUMBERS. 


7  times    0  are    0 

9  times    0  are      0 

11  times    0  are      0 

7  times     1  are    7 

9  times     1  are       9 

11  times     1  are     11 

7  times     2  are  14 

9  times    2  are     18 

11  times    2  are    22 

7  times     3  are  21 

9  times    3  are    27 

11  times    3  are    33 

7  times    4  are  28 

9  times     4  are     36 

11  times    4  are     44 

7  times     5  are  85 

9  times     5  are    45 

11  times    5  are    55 

7  times     6  are  42  1    9  times     G  are    54 

11  times    6  are     66 

7  times     7  are  49  }    9  times    7  are    68 

11  times    7  are    77 

7  times    8  are  56 

9  times     8  are     72 

11  times    8  are     88 

7  times    9  are  63 

9  times    9  are    81 

11  times    9  are    99 

7  times  10  are  70 

9  times  10  are    90 

11  times  10  are  110 

7  times  11  are  77 

9  times  11  are    99 

11  timfes  11  are  121 

7  times  12  are  84 

9  times  12  are  108 

11  times  12  are  132 

8  times     0  are    0 

10  times     0  are      0 

12  times    0  are      0 

8  times     1  are    8 

10  times    1  are    10 

12  times     1  are     12 

8  times    2  are  16 

10  times    2  are    20 

12  times     2  are    24 

8  times    3  are  24 

10  times    3  are    30 

12  times     3  are    36 

8  times    4  are  32 

^10  times    4  are    40 

12  times    4  are     48 

8  times    5  are  40 

"lO  times    5  are    50 

12  times    5  are     60 

8  times    6  are  48 

10  times    6  are    60 

12  times    6  are     72 

8  times    7  are  56 

10  times    7  are    70 

12  times     7  are     84 

8  times    8  are  64 

10  times    8  are    80 

12  times     8  are     96 

8  times     9  are  72 

10  times    9  are    90 

-12  times     9  are  108 

8  times  10  are  80 

10  times  10  are  100 

12  times  10  are  12G 

8  times  11  are  88 

10  times  11  are  110 

12  times  11  are  132  j 

8  times  12  are  96 

10  times  12  are  120 

12  times  12  are  144 

4.  What  is  the  cost  of  6  yards  of  ribbon  at  7  cents  a  yard  ? 

ANALYSIS. — Six  yards  of  ribbon  will  cost  6  times  as  much  as 
1  yard.  Since  1  yard  costs  7  cents,  6  yards  will  cost  6  times 
7  cents,  which  are  42  cents. 

Let  the  pupil  analyze  every  question  in  a  similar  manner. 

5.  What  will  8  yards  of  muslin  cost  at  9  cents  a  yard  ? 

6.  What  will  9  pounds  of  sugar  cost  at  9  cents  a  pound  ? 

7.  What  is  the  cost  of  7  pounds  of  butter  at  12  cents  a 
pound  ? 

8.  What  is  the  cost  of  12  pounds  of  tea  at  6  shillings  a 
pound  ? 

9.  What  is  the  cosf  of  12  pounds  of  coffee  at  9  cents  a 
pound  ? 

10.  What  is  the  cost  of  11  yards  of  cloth  at  6  dollars  a 
yard  ? 

11.  What  is  the  cost  of  9  books  at  11  cents  each  ? 


44  MULTIPLICATION. 

12.  What  is  the  cost  of  12  pencils  at  8  cents  apiece  ? 

13.  What  is  the  cost  of  10  pairs  of  shoes'  at  2  dollars  a 
pair  ? 

14.  What  is  the  cost  of  12  pairs  of  stockings  at  3  shillings 
a  pair  ? 

PRINCIPLES    AND    EXAMPLES 

45.  Let  it  bo  required  to  multiply  4  by  3,  and  also  to  mul- 
tiply 5  by  3. 


OPERATION. 


li 

-t-3        -£3 


i  i    i 

4  X3  =  1     4 


12     Product. 


OPERATION. 


15     Product. 


From  the  first  of  these  examples  we  see,  that  the  product 
of  4  multiplied  by  3,  is  12,  the  number  which  arises  from 
taking  4,  3  times  ;  and  that  the  product  of  5  by  3  is  equal  to 
15,  the  number  which  arises  from  taking  5,  three  times : 
hence, 

MULTIPLICATION  is  the  operation  of  taking  one  number  as 
many  times  as  there  are  units  in  another. 

The  number  to  be  taken  is  called  the  multiplicand. 

The  number  denoting  how  many  times  the  multiplicand  is 
taken,  is  called  the  multiplier. 

The  result  of  the  operation  is  called  the  product. 

The  multiplicand  and  multiplier  are  called  factors,  or  pro- 
ducers of  the  product. 

46.  We  also  see,  from  the  above  examples,  that  4  taken 
3  times,  gives  the  same  result  as  is  obtained  by  adding  three 
4's  together ;  and  that  5  taken  3  times  gives  the  same  result 
as  is  obtained  by  adding  three  5's  together  :  hence, 

45.  What  is  Multiplication  ?    What  is  the  number  called  which  is  to 
be  taken?     What  does   the  multiplier  denote?     What  is   the  result 
called  ?    What  are  the  multiplier  and  multiplicand  called  ? 

46.  What  is  4  multiplied  by  3  equal  to  ?    What  is  5  multiplied  by  3 
equal  to  ?    How  then  may  multiplication  be  di -lined  ? 


SIMPLE   NUMBERS.  45 

MULTIPLICATION  is  a  short  method  of  addition. 

47.  The  sign  x,  placed  between  two  numbers,  denotes 
that  they  are  to  be  multiplied  together.  It  is  called  the  sign 
of  multiplication.  Also,  ( 4  -f  3 )  x  5,  denotes  that  the  sum  of  4 
and  3  is  to  be  multiplied  by  5. 


9x8=  72. 
Ix2x  3=  6. 
Ix4x  5=  20. 
2x6x  5=  60. 
3  x  4  x  9  =  how  many  ? 
4x3x11=  how  many  ? 

5  x  2  x    9  =  how  many  ? 

6  x  2  x    5  =  how  many  ? 


7  x  8  =  how  many  ? 
1  x  6  x  9  =  how  many  ? 
1  x  9  x  12=  how  many  ? 


5  x  2  x  11=  how  many  ? 
7  x  1  x  12=  how  many  ? 
9  x  1  x  9=  how  many  ? 

11  x  1  x    7  =  how  many  ? 

12  x  1  x    5=  how  many  ? 


NOTE. — There  are  three  parts  in  every  operation  of  multiplica- 
tion. First,  the  multiplicand:  second,  the  multiplier:  and  third, 
the  product. 

48.  The  product  of  two  factors  is  the  same,  whichever  be 
taken  for  the  multiplier.  /  ( 

For,  let  it  be  required  to  multiply  5  by  3. 

OPERATION. 

ANALYSIS.— Place    as    many  1's   in  a                        ,5 
horizontal  row  as  there  are  units  in  the  , 


multiplicand,  and  make  as  many  rows  as  Mill! 

there  are  units  in  the    multiplier  :    the  \  | 

product  is  equal  to  the  number  of  1's  in  o  -j  1 

one  row  taken  as  many  times  as  there  are  (  1       11      1       1 

rows  :  that  is,  to  0  x  3=15.  JT 

But  if  we  consider  the  number  of  1  s  in  a  vertical  row  to  be 
the  multiplicand,  and  the  number  of  vertical  rows  the  multiplier, 
the  product  will  be  equal  to  the  number  of  1's  in  a  vertical  row 
taken  as  many  times  as  there  are  vertical  rows ;  that  is,  3  x  5=15  : 
and,  as  the  same  may  be  shown  for  any  two  numbers, 

The  product  of  two  factors  is  the  same  whichever  factor 
is  used  as  the  multiplier. 

47.  What  is  the  sign  of  multiplication  ? 

NOTE. — How  many  parts  are  there  in  any  operation  of  multiplica- 
tion ?  What  are  they  ? 

48.  What  is  the  product  of  3  by  4  ?    Of  4  by  3  ?    Is  the  product 
altered  by  changing  the  order  of  the  factors  ? 


4:6  MULTIPLICATION. 


EXAMPLES. 

3x7  =  7x3  =  21:     also,  6x3  =  3x6=18. 

9  x  5=5  x  9=45  :     also,  8  x  6=6  x  8  =  48. 

and,  8x7  =  7x8=56:     also,  5x7  =  7x5  =  35. 

-  49.    When  the  multiplier  does  not  exceed  12 
1.  Let  it  be  required  to  multiply  236  by  4. 

ANALYSIS.  —  It  is  required  to  take  230  4    OPERATION. 
times.     If  the  entire  number  is  taken  4  times,      236 
each  order  of  units  must  be  taken  4  times  :          4. 
hence,  the  product  must  contain  24  units,  12     - 
tens,  and  8  hundreds  ;  therefore,  the  product        24    units. 
is  944.  12      tens. 

It  is  seen,  from  the  preceding  analysis.     8  _    hundreds. 
that,  944"  Product. 

1.  If  units  be  multiplied  by  units,  the  unit  of  the  product 
will  be  1. 

2.  If  tens  be  multiplied  by  units,  the  unit  of  the  product 
unit  be  1  ten. 

3.  If  hundreds  be  multiplied  by  units,  the  unit  of  the 
product  will  be  1  hundred  ;  and  so  on  : 

And  since  the  product  of  the  factors  is  the  same  whichever 
is  taken  for  the  multiplier  (Art.  48),  it  follows  that, 

4.  If  units  of  the  first  order  be  multiplied  by  units  of  a 
higher  order,  the  units  of  the  product  will  be  the  mme  as 
that  of  the  higher  order.  / 

The  operation  in  the  last  example  may  be  performed  ia 
another  way,  thus  : 

ANALYSIS.  —  Say  4  times  6  are  24  :  set  down  the     OPERATION. 
4,  and  then  say,  4  times  3  are  12,  and  2  to  carry  236 

are  14  ;  set  down  the  4,  and  then  say,  4  times  2  are  4 

8,  and  1  to  carry  are  9.     Set  down  the  9,  and  the 
product  is  944  as  before. 

The  method  of  carrying  is  the  same  as  in  addition. 


(1.)         (2.)  (3.)  (4.) 

867901  278904  678741  3021945 

1  2  .  3  _J 

867901  12087780 


SIMPLE   NUMBERS.  47 

(5.)                          (6)  (7.)  (8.) 

28432  82798  6789  49604 

8  _  _9  11  _  12 

227456  595248 

9.  A  merchant  sold  104  yards  of  cotton  sheeting  at  9  cents 
a  yard  :  what  did  he  receive  for  it  ? 

10.  A  farmer  sold  309  sheep  at  four  dollars  apiece  :  how 
much  did  he  receive  ? 

11.  Mrs.  Simpkins  purchased  149  yards  of  table  linen  at 
two  dollars  a  yard  :  how  much  did  she  pay  for  it  ? 

12.  What  is  the  cost   of  2974  pine-apples  at  12  cents 
apiece  ? 

13.  What  is  the  cost  of  4073  yards  of  cloth  at  7  dollars 
a  yard  ? 

14.  What  is  the  cost  of  a  drove  of  598  hogs  at  11  dollars 
apiece  ? 

READING    RESULTS. 

50.  Spelling,  IP  multiplication,  is  naming  the  two  factors 
which  produce  the  product,  as  well  as  the  words  which  in- 
dicate the  operation  ;  whilst  the  reading  consists  in  naming 
only  the  word  which  expresses  the  final  result. 

ANALYSIS.—  In  multiplying  8325  by  6,  we  say,  OPERATION. 
6  times  5  are  30  ;  then,  6  times  2  are  12  and  3  to  8325 

carry  are  15  ;  6  times  3  are  18  and  1  to  carry  are  6 

19  ;  C  times  8  are  48  and  1  to  carry  are  49. 


This  is  the  spelling.  The  reading  consists  in  pronouncing 
only  each  final  word  which  denotes  the  result  of  an  operation 
thus  :  thirty,  fifteen,  nineteen,  forty-nine. 

With  a  little  practice,  the  pupils  will  perform  the  operations 
mentally,  and  read  with  great  facility,  either  separately  or  in 
concert  in  classes. 

51.  When  the  multiplier  exceeds  12. 
i.  Multiply  8204  by  603. 


49.  Explain  the  multiplication  of  336  by  4  ?    What  principles  are 
established  by  this  operation  ? 

50.  Explain  the  manner  of  reading  the  results  in  the  operations  of 
multiplication  ? 

51.  Give  the  rule  for  multiplication 


48  MULTIPLICATION. 


ANALYSIS.—  The  multiplicand  is  to  be  taken  603  R90  1 
times.     Taking  it  3  times  we  obtain  24612. 

When  we  come  to  take  it  6  hundreds  times,  the  _  5__ 

lowest  order  of  units  will  be  hundreds:    hence,  4,  24612 

the  first  figure  of  the  product,  must  be  written  in  10091 
the  third  place. 

4947012 

NOTE.  —  The  product  obtained  by  multiplying  by  a  single  figure 
of  the  multiplier,  is  called  a  partial  product.  In  the  above  ex- 
ample there  are  two  partial  products,  24612  and  49224.  The 
sum  of  the  partial  products  is  equal  to  the  result  or  product  sought  : 
hence,  the  following 

RULE  —  I.  Write  the  multiplier  under  the  'multiplicand, 
placing  units  of  the  same  order  in  the  same  column. 

II.  Beginning  ivith  the  units'  figure,  multiply  the  entire 
multiplicand  by  each  figure  of  the  multiplier,  observing  to 
write  the  first  figure  of  each  partial  product  directly  under 
its  multiplier.  , 

III.  Add  the  partial  products  and  their  sum  will  be 
the  product  sought. 

PROOF. 

52.  Write  the  multiplicand  in  the  place  of  the  multiplier 
and  find  the  product  as  before.  If  the  two  products  are  the 
same,  the  work  is  supposed  to  be  right. 

NOTE.  —  This  proof  depends  on  the  principle  that  the  product  of 
two  numbers  is  the  same  whichever  is  taken  for  the  multiplicand 
(Art.  48)  ;  and  also  on  the  fact,  that  the  same  error  would  not  be 
likely  to  occur  in  both  operations. 

EXAMPLES. 

1.  Multiply  354  by  267. 


Multiplicand, 
Multiplier, 

Product, 

OPERATION. 

354 
267 

"2478 
2124 

708 

PROOF. 

267 
354 

1068 
1335 
801 

94518 

94518 

52.  How  do  you  prove  multiplication  ? 


SIMPLE   NUMBERS. 


2.  Multiply  365  by  84  ;  also  37864  by  209. 


(2.) 
Multiplicand,       365 
Multiplier,             84 

(3.) 
37864 
209 

(4.) 
34293 

74 

(5.) 
47042 
91 

1460 
2920 

Product, 


30660 


4280822 


(6.) 
46834 

679084 

(8.) 
1098731 

(9.) 
8971432 

406 

126 

1987 

10471 

19014604 

10.  Multiply  12345678  by  32. 

11.  Multiply  9378964  £y  42. 

12.  Multiply  1345894  by  49. 

13.  Multiply  576784  by  64. 


14.  Multiply  596875  by  144. 

15.  Multiply  46123101  by  72. 

16.  Multiply  6185720  by  132. 

17.  Multiply  7 18328  by  96. 


18.  Multiply  five  thousand  nine  hundred  and  si^ty-five,  by 
six  thousand  and  nine. 

19.  Multiply  eight  hundred  and  seventy  thousand  six  hun- 
dred and  fifty-one,  by  three  hundred  and  seven  thousand  and 
four. 

20.  Multiply  four  hundred  and  sixty-two  thousand  six  hun- 
dred and  nine,  by  itself. 

21.  Multiply  eight  hundred  and  forty-nine  million,  six  hun- 
dred and  seven  thousand,  three  hundred  and  six,  by  nine 
hundred  thousand,  two  hundred  and  four. 


22.  Multiply  679534  by  9185. 

23.  Multiply  86972  by  1208. 

24.  Multiply  1055054  by  570. 

25.  Multiply  538362  by  9258. 


26.  Multiply  50406  by  8050. 

27.  Multiply  523972  by  1527. 

28.  Multiply  760184  by  1615. 

29.  Multiply  105070  by  3145. 


CONTRACTIONS  IN  MULTIPLICATION. 

53.  Contractions  in  multiplication  are  short  methods  of 
finding  the  product  when  the  multiplier  is  a  composite  num- 
ber. 


53.  What  are  contractions  in  multiplication  ? 
4 


50  MULTIPLICATION. 

CASE    I. 

Of  Components  or  Factors. 

54.  A  composite  number  is  one  that  may  be  produced  by 
the  multiplication  of  two  or  more  numbers,  which  are  called 
components  or  factors. 

Thus,  2  x  3=6.  Hence,  6  is  the  composite  number,  and  2 
and  3  are  its  components  or  factors. 

The  number,  16=8x2:  here  16  is  a  composite  number, 
and  8  and  2  are  the  factors.  But  since  4  x4=16,  we  may 
also  regard  4  and  4  as  factors  of  16. 

Again,  16=8x2,  and  8  =  4x9  =  2x2x2:  hence, 
16=2x2x2x2:  therefore,  16  has  also  four  equal  factors. 

1.  What  are  the  factors  of  8  ?  of  9  ?  of  10  ?  of  12?  of  14? 
of  18  ?  of  24  ? 

2.  What  are  the  factors  of  20  ?  of  21  ?  of  22  ?  of  26  ;  of 
25?  of  30? 

3.  What  are  the  factors  of  36  ?  of  42  ?  of  44  ?  of  49  ?  of 
56?  of  64?  of  72? 

4.  Let  it  be  required  to  multiply  8  by  the  composite  num- 
ber 6,  of  which  the  factors  are  2  and  3. 


1  1  1  1  1  1  1  1(0VQ  1* 
1111111  l|2X8=:1* 
1  1  1  1  1  1  1  *  ' 


50  |q     (1     1     1     1     1     1     1     l|2  48         24 

'    -h     1     1     1     1     1     1     1) 9  2 

(11111111)  48 

If  we  write  6  horizontal  lines  with  8  units  in  each,  it  is 
evident  that  the  product  of  8  x  6=48  will  express  the  num- 
ber of  units  in  all  the  lines. 

Let  us  first  connect  the  lines  in  sets  of  two  each,  as  at  the 
right ;  the  number  of  units  in  each  set  will  then  be  expressed 
by  8  x  2=16.  But  there  are  3  sets  ;  hence,  the  number  of 
units  in  all  the  sets  is  16  x  3  =  48. 

54.  What  is  a  composite  number  ?  Is  6  a  composite  number  ?  What 
are  its  components  or  factors  ?  What  are  the  factors  of  the  composite 
number  16  ?  What  are  the  factors  of  the  composite  number  12  ?  How 
do  you  multiply  when  the  multiplier  is  a  composite  number? 


SIMPLE   NUMBERS  51 

Again,  if  we  divide  the  lines  into  sets  of  3  each,  as  at  the 
left,  the'  number  of  units  in  each  set  will  be  equal  to 
8x's=24,  and  since  there  are  two  sets,  the  whole  number 
of  units  will  be  expressed  by24x2=48. 

Since  the  product  of  either  two  of  the  three  factors  8,  3  and 
2,  win  be  the  same  whichever  be  taken  for  the  multiplier 
(48),  and  since  the  same  principle  will  apply  to  that  product 
and  the  other  factor,  as  well  as  to  any  additional  factor,  if 
introduced,  it  follows  that, 

The  product  of  any  number  of  factors  will  be  the  same 
in  whatever  order  they  are  multiplied :  hence,  the  following 

RULE. — I.   Separate  the  composite  number  into  its  factors. 

II.  Multiply  the  multiplicand  and  the  partial  products 
by  the  factors,  in  succession,  and  the  last  product  mill  be  the 
entire  product  sought. 

EXAMPLES. 

1.  Multiply  327  by  12. 

The  factors  of  12  are  2  and  6  ;  they  are  also  3  and  4  ;  or 
fhey  are  3,  2  and  2. 

For,  2x6  =  12,  3x4  =  12,  and  3x2x2  =  12. 


2.  Multiply  5709  by  48. 

3.  Multiply  342516  by  56. 

4.  Multiply  209402  by  72. 


5.  Multiply  937387  by  54. 

6.  Multiply  91738  by  81. 

7.  Multiply  3842  by  144. 


CASE    II. 

55.  When  the  multiplier  is  1,  with  any  number  of  ci- 
phers annexed,  as  10,  100,  1000,  &c. 

Placing  a  cipher  on  the  right  of  a  number,  is  called  an- 
nexing it.  Annexing  one  cipher  increases  the  unit  of  each 
place  ten  times  :  that  is,  it  changes  units  into  tens,  tens  into 
hundreds,  hundreds  into  thousands,  &c. ;  and  therefore  in- 
creases the  number  ten  times. 

Thus,  the  number  5  is  increased  ten  times  by  annexing  one 
cipher,  which  makes  it  50.  The  annexing  of  two  ciphers 

55.  If  yon  place  one  cipher  on  the  right  of  a  number,  what  effect  has 
it  on  its  value  ?  If  you  place  two,  what  effect  has  it  ?  If  you  place 
three  ?  How  much  will  each  increase  it  ?  How  do  you  multiply  by 
10,  100,  1000,  &c  ? 


52  MULTIPLICATION. 

increases  a  number  one  hundred  times  ;  the  annexing  of  three 
ciphers,  a  thousand  times,  &c. :  hence  the  following 

RULE. — Annex  to  the  multiplicand  as  many  ciphers  as 
there  are  in  the  multiplier,  and  the  number  so  formed  will 
be  the  required  product. 


EXAMPLES. 


1.  Multiply  254  by  10. 

2.  Multiply  648  by  100. 

3.  Multiply  7987  by  1000. 

4.  Multiply  9840  by  10000. 


5.  Multiply  3750  by  100. 

6.  Multiply  6704  by  10000. 

7.  Multiply  2141  by  100. 

8.  Multiply  872  by  100000. 


CASE    III. 

56.  When  there  are  ciphers  on  the  right  hand  of  one  or 
both  of  the  factors. 

In  this  case  each  number  may  be  regarded  as  a  composite 
number,  of  which  the  significant  figures  are  one  factor,  and 
1,  with  the  requisite  number  of  ciphers  annexed,  the  other. 

1.  Let  it  be  required  to  multiply  3200  by  800- 

OPERATION. 

3200=32  x  100  ;  and  800=8  x  100  ; 
Then,  3200  x  800  =  32  x  100  x  8  x  100 
=  32x8x100x100 
=  2560000. 

Hence,  we  have  the  following 

RULE. — Omit  the  ciphers  and  multiply  the  significant 
figures :  then  place  as  many  ciphers  at  the  right  hand  of 
the  product  as  there  are  in  both  factors. 

EXAMPLES. 

(1.)  (2.)  (3.) 

76400          7532000          416000 
24  580  357000 


1«33600  148512000000 


4.  4871000x270000. 

5.  296200x875000. 

6.  3456789x567090. 


7.  21200x70. 

8.  359260x304000. 

9.  7496430x695000. 


SIMPLE   NUMBERS.  53 

APPLICATIONS    IN    MULTIPLICATION. 

57.  The  analysis  of  a  practical  question,  in  Multiplication, 
requires  that  the  multiplier  be  an  abstract  number  ;  and  then 
the  unit  of  the  product  will  be  the  same  as  the  unit  of  the 
multiplicand. 

Thus,  what  will  5  yards  of  cloth  cost  at  7  dollars  a  yard  ? 

ANALYSIS. — Five  yards  of  cloth  will  cost  5  times  as  much  as 
1  yard.  Since  1  yard  of  cloth  costs  7  dollars,  5  yards  will  cost 
5  times  7  dollars,  which  are  35  dollars. 

The  cost  of  any  number  of  things  is  equal  to  the  price 
of  a  single  thing  multiplied  by  the  number. 

But  we  have  seen  that  the  product  of  two  numbers  will  be 
the  same,  (that  is,  will  contain  the  same  number  of  units) 
whichever  be  taken  for  the  multiplicand  (Art.  48).  Hence, 
in  practice,  we  may  multiply  the  two  factors  together,  taking 
either  for  the  multiplier,  and  than  assign  the  proper  unit  to 
the  product,  We  generally  take  the  least  number  for  the 
multiplier. 

QUESTIONS    FOR    PRACTICE. 

1.  There  are  ten  bags  of  coffee,  each  containing  48  pounds  : 
how  much  coffee  is  there  in  all  the  bags  ? 

2.  There  are  20  pieces  of  cloth,  each  containing  37  yards, 
and  49  other  pieces,  each  containing  75  yards :  how  many 
yards  of  cloth  are  there  in  all  the  pieces  ? 

3.  There  are  24  hours  in  a  day,  and  7  days  in  a  week : 
how  many  hours  in  a  week  ? 

4.  A  merchant  buys  a  piece  of  cloth  containing  97  yards, 
at  3  dollars  a  yard :  what  does  the  piece  cost  him  ? 

5.  A  farmer  bought  a  farm  containing  10  fields  ;  three  of 
the  fields  contained  9  acres  each  ;  three  other  of  the  fields 
12  acres  each ;  and  the  remaining  4  fields  each  15  acres : 
how  many  acres  were  there  in  the  farm,  and  how  much  did 
the  whole  cost  at  18  dollars  an  acre? 

6.  Suppose  a  man  were  to  travel  32  miles  a  day  :  how  far 
would  he  travel  in  365  days  ? 

56.  When  there  are  ciphers  on  the  right  hand  of  one  or  both  the  fac- 
tors, how  do  you  multiply  ? 

57.  What  does  the  analysis  of  a  practical  question  require?    How  do 
you  find  the  cost  of  a  single  thing  ?    How  may  it  be  done  in  practice  ? 


54  MULTIPLICATION. 

7.  A  merchant  bought  49  hogsheads  of  molasses,  each 
containing  63  gallons  :  how  many  gallons  of  molasses  were 
there  in  the  parcel  ? 

8.  In  a  certain  city  there  are  3751  houses.     If  each  house 
on  an  average  contains  5  persons,  how  many  inhabitants  are 
there  in  the  city  ? 

9.  If  a  regiment  of  soldiers  contains  1128  men,  how  many 
men  are  there  in  an  army  of  106  regiments  ? 

10.  If  786  yards  of  cloth  can  be  made  in  one  day,  how 
many  yards  can  be  made  in  1252  days  ? 

11.  If  30009  cents  are  paid  for  one  man's  labor  on  a  rail- 
road for  1  year,  how  many  cents  would  be  paid  to  814  men, 
each  man  receiving  the  same  wages  ? 

12.  There  are  320  rods  in  a  mile;  how  many  rods  are 
there  in  the  distance  from  St.  Louis  to  New  Orleans,  wind. 
is  1092  miles  ? 

13.  Suppose  a  book  to  contain  470  pages,  45  lines  on  each 
page,  and  50  letters  in  each  line  :  how  many  letters  in  the 
book? 

14.  Supposing  a  crew  of  250  men  to  have  provisions  for 
30  days,  allowing  each  man  20  ounces  a  day :  how  many 
ounces  have  they  ? 

15.  There  are  350  rows  of  trees  in  a  large  orchard,  125 
trees  in  each  row,  and  3000  apples  on  each  tree :  how  man}1 
apples  in  the  orchard  ? 

16.  What  is  the  cost  of  7585  barrels  of  flour  at  7  dollars  a 
barrel  ? 

17.  If  a  railroad  car  goes  27  miles  an  hour,  how  far  will 
it  run  in  3  days,  running  20  hours  each  day  ?    How  far  would 
it  run  if  its  rate  were  37  miles  an  hour  ? 

18.  If  1327  barrels  of  flour  will  feed  the  inhabitants  of  a 
city  for  1  day,  how  many  barrels  will  supply  them  for  2 
years  ? 

19.  A  regiment  of  men  contains  10  companies,  each  com- 
pany 8  platoons,  and  each  platoon  34  men :  how  many  men 
in  the  regiment  ? 

20.  Two  persons  start  from  the  same  place  and  travel  in 
the  same  direction :  one  travels  at  the  rate  of  6  miles  an 
hour,  the  other  at  the  rate  of  9  miles  an  hour.     If  they  travel 
8  hours  a  day,  how  far  will  they  be  apart  at  the  end  of  17 
days  ?     How  far  if  they  travel  in  opposite  directions  ? 


SIMPLE   NUMBERS.  55 

21.  The  Erie  railroad  is  about  425  miles  long,  and  cost  65 
thousand  dollars  a  mile :  what  was  the  entire  cost  of  con- 
struction ? 

22.  A  drover  bought  106  oxen  at  35  dollars  a  head  ;  it  cost 
him  6  dollars  a  head  to  get  them  to  market,  where  he  sold 
them  at  47  dollars  ;  did  he  make  or  lose,  and  how  much  ? 

23.  The    great   Illinois   Central   Railroad    reaches   from 
Chicago  to  the  mouth  of  the  .Ohio  river,  815  miles :  it  cost 
23500  dollars  a  mile  :  what  was  its  entire  cost  ? 

24.  Mr.  Denning's  orchard  is  square  and  contains  36  trees 
in  a  row  :  each  tree  yields  4  barrels  of  apples  which  he  sells 
for  2  dollars  a  barrel :  how  much  does  he  get  for  his  crop  ? 

BILLS    OF    PARCELS. 

58.  When  a  person  sells  goods  he  generally  gives  with 
them  a  bill,  showing  the  amount  charged  for  them,  and 
acknowledging  the  receipt  of  the  money  paid ;  such  bills  are 
called  Mills  of  Parcels. 

New  York,  Oct.  1,  1854. 

25  James  Johnson,  Bought  of  W.  Smith. 
4  Chests  of  tea,  of  45  pounds  each,  at  1  doll,  a  pound. 

3  Firkins  of  butter  at  1 7  dolls,  per  firkiu 

4  Boxes  of  raisins  at  3  dolls,  per  box    ... 
36  Bags  of  coffee  at  16  dolls,  each 

14  Hogsheads  of  molasses  at  28  dolls,  each     - 

Amount,  dollars. 

Received  the  amount  in  full.  W.  Smith 

Hartford,  Nov.  1,  1854. 

26  James  Hughes,  Bought  of  W.  Jones. 

27  Bags  of  coffee  at  14  dollars  per  bag  - 
18  Chests  of  tea  at  25  dolls,  per  chest     - 
75  Barrels  of  shad  at  9  dolls,  per  barrel 

87  Barrels  of  mackerel  at  8  dolls,  per  barrel  - 

67  Cheeses  at  2  dolls,  each     - 

59  Hogsheads  of  molasses  at  29  dolls,  per  hogshead, 

Amount,  dollars. 

Received  the  amount  in  full,  for  W.  Jones, 

per  James  Cross. 

58.  What  are  bills  of  parcels  ? 


56 


DIVISION. 


DIVISION. 

59.  1.  How  many  1's  are  there  in  1  ?     How  many  in  2  ? 
In  3  ?     In  4  ?     In  5  ? 

2.  How  many  2's  are  there  in  2  ?  2  in  2  how  many  times  ? 
2  in  4  how  many  times  ?  2  in  6  how  many  times  ?     In  8  ? 

3,  How  many  3's  in  6  ?    3  in   6  how  many  times  ?    3  in 
9?    3  in  12?    3  in  15?    3  in  18  ? 

DIVISION  TABLE. 


1  in    1     1  time 
1  in    2    2  times 
1  in    3    3  times 
1  in    4    4  times 
1  in    5    5  times 
1  in     6    6  times 
1  in    7    7  times 
1  in    8    8  times 
1  in    9    9  times 

5  in    5     1  time 
5  in  10    2  times 
5  in  15    3  times 
5  in  20    4  times 
5  in  25     5  times 
5  in  30    6  times 
5  in  35     7  times 
5  in  40    8  times 
5  in  45     9  times 

9  in      91  time 
9  in     18    2  times 
9  in    27    3  times 
9  in     36    4  times 
9  in    45    5  times 
9  in     54    6  times 
9  in     63     7  times 
9  in     72     8  times 
9  in    81     9  times 

2  in    2     1  time 
2  in    4    2  times 
2  in    6     3  times 
2  in    8    4  times 
2  in  10    5  times 
2  in  12    6  times 
2  in  14    7  times 
2  in  16     8  times 
2  in  18    9  times 

6  in     6     1  time 
6  in  12     2  times 
6  in  18    3  times 
6  in  24    4  times 
6  in  30    5  times 
6  in  36    6  times 
6  in  42     7  times 
6  in  48     8  times 
6  in  54     9  times 

10  in     10     1  time 
10  in     20     2  times 
JO  in     30    3  times 
10  in     40    4  times 
10  in    50    5  times 
10  in     60     6  times 
10  in     70    7  times 
10  in     80    8  times 
10  in     90    9  times 

3  in    3     1  time 
3  in    6    2  times 
3  in    9     3  times 
3  in  12    4  times 
3  in  15    5  times 
3  in  18    6  times 
3  in  21     7  times 
3  in  24    8  times 
3  in  27    9  times 

7  in     7     1  time 
7  in  14    2  times 
7  in  21     3  times 
7  in  28    4  times 
7  in  35    5  times 
7  in  42     6  times 
7  in  49     7  times 
7  in  56    8  times 
7  in  63    9  times 

11  in     11     1  time 
11  in     22    2  times 
11  in     33    3  times 
11  in     44    4  times 
11  in     55    5  times 
11  in     66    6  times 
11  in     77    7  times 
11  in     88    8  times 
11  in     99    9  times 

4  in    41  time 
4  in    8    2  times 
4  in  12    3  times 
4  in  16    4  times 
4  in  20    5  times 
4  in  24    6  times 
4  in  28    7  times 
4  in  32    8  times 
4  in  36    9  times 

8  in    8     1  time 
8  in  16    2  times 
8  in  24    3  times 
8  in  32    4  times 
8  in  40    5  times 
8  in  48    6  times 
8  in  56    7  times 
8  in  64    8  times 
8  in  72    9  times 

12  in     12  *1  time 
12  in     24     2  times 
12  in     36    3  times 
12  in     48    4  times 
12  in    60    5  times 
12  in     72    6  times 
12  in    84    7  times 
12  in     96    8  times 
12  in  108    9  times 

SIMPLE   NUMBERS.  57 


QUESTIONS. 

1.  If  12  apples  be  equally  divided  among  4  boys,  how 
many  will  each  have  ? 

ANALYSIS. — Since  12  apples  are  to  be  divided  equally  among 
4  boys,  one  boy  will  have  as  many  apples  as  4  is  contained  times 
in  12,  which  is  3. 

2.  If  24  peaches  be  equally  divided  among  6  boys,  how 
many  will  each  have  ?     How  many  times  is  6  contained  in 
24? 

3.  A  man  has  32  miles  to  walk,  and  can  travel  4  miles  an 
hour,  how  many  hours  will  it  take  him  ? 

4.  How  many  yards  of  cloth,  at  3  dollars  a  yard,  can  you 
buy  for  24  dollars  ? 

ANALYSIS. — Since  the  cloth  is  3  dollars  a  yard,  you  can  buy  as 
many  yards  as  3  is  contained  times  in  24,  which  is  8 :  therefore, 
you  can  buy  8  yards. 

5.  How  many  oranges  at  6  cents  apiece  can  you  buy  for 
42  cents  ? 

6.  How  many  pine-apples  at  12  cents  apiece  can  you  buy 
for  132  cents  ? 

7.  A  farmer  pays  28  dollars  for  7  sheep :  how  much  is 
that  apiece  ? 

ANALYSIS. — Since  7  sheep  cost  28  dollars,  one  sheep  will  cost  as 
many  dollars  as  7  is  contained  times  in  28,  which  is  4 ;  therefore, 
each  sheep  will  cost  4  dollars. 

8.  If  12  yards  of  muslin  cost  96  cents,  how  much  does 
1  yard  cost  ? 

9.  How  many  lead  pencils  could  you  buy  for  42  cents,  if 
they  cost  6  cents  apiece  ? 

10.  How  many  oranges  could  you  buy  for  72  cents,  if  they 
cost  6  cents  apiece  ? 

11.  A  trader  wishes  to  pack  64  hats  in  boxes,  and  can  put 
but  8  hats  in  a  box  :  how  many  boxes  does  he  want  ? 

12.  If  a  man  can  build  7  rods  of  fence  in  a  day,  how  long 
will  it  take  him  to  build  7  7  rods  ? 

13.  If  a  man  pays  56  dollars  for  seven  yards  of  cloth,  how 
much  is  that  a  yard  ? 


58 


DIVISION. 


14.  Twelve  men  receive  108  dollars  for  doing  a  piece  of 
work  :  how  much  does  each  one  receive  ? 

15.  A  merchant  has  144  dollars  with  which  he  is  going  to 
buy  cloth  at  12  dollars  a  yard  ;  how  many  yards  can  he  pur- 
chase ? 

16.  James  is  to  learn  forty-two  verses  of  Scripture  in  a 
week  :  how  many  must  he  learn  each  day  ? 

17.  How  many  times  is  4  contained  in  50,  and  how  many 
over? 

PRINCIPLES    AND    EXAMPLES. 


60.  1.  Let  it  be  required  to  divide  86  by  2. 

Set  down  the  number  to  be  divided  and  write 
the  other  number  on  the  left,  drawing  a  curved 
line  between  them.  Now  there  are  8  tens  and 
6  units  to  be  divided  by  2.  We  say,  2  in  8,  4 
times,  which  being  tens,  we  write  it  in  the  tens' 
place.  We  then  say,  2  in  6,  3  times,  which 
being  units,  are  written  in  the  units'  place. 
The  result,  which  is  called  a  quotient,  is  there- 
fore, 4  tens  and  3  units,  or  43. 

2.  Let  it  be  required  to  divide  729  by  3. 


OPERATION. 


2)  86 

43  quotie't. 


ANALYSIS.  —  We  say,  3  in  7,  2  times  and  1  over.    OPERATION. 


Set  down  the  2,  which  are  hundreds,  under  the  7. 
But  of  the  7  hundreds  there  is  1  hundred,  or  10  tens, 
not  yet  divided.  We  put  the  10  tens  with  the  2 


3)729 
1243 


tens,  making  12  tens,  and  then  say,  3  in  12,  4  times,  and  write  the 
4  of  the  quotient  in  the  tens'  place ;  then  say,  3  in  9,  3  times. 
The  quotient,  therefore,  is  243. 

3.  Let  it  be  required  to  divide  466  by  8. 

ANALYSIS.— We  first  divide  the  46  tens 
by  8,  giving  a  quotient  of  5  tens,  and  6  tens 
over.  These  6  tens  are  equal  to  60  units, 
to  which  we  add  the  6  in  the  units'  place. 
We  then  say,  8  in  66,  8  times  and  2  over ; 
hence,  the  quotient  is  58,  and  2  over,  which 
we  caU  a  remainder.  This  remainder  is 
written  after  the  last  quotient  figure,  and 
the  8  paced  under  it;  the  quotient  is  read, 
58  and  2  divided  by  8- 


OPERATION. 

8)466 

58-2  remain. 


58f  quotient. 


50.  Ex.  1.— When  you  divide  8  tons*  by  2,  is  the  unit  of  the  quotient 
tens  or  units  ?    When  6  units  are  divided  by  2,  what  is  the  unit  ? 


SIMPLE   NUMBERS.  59 

ANALYSIS.— In  the  first  example  86  is  divided  into  2  equal  parts, 
and  the  quotient  43  is  one  of  the  parts.  If  one  of  the  equal  parts 
be  multiplied  by  the  number  of  parts  2,  the  product  will  be  86,  the 
number  divided. 

In  the  third  example  466  is  divided  into  8  equal  parts,  and  two 
units  remain  that  are  not  divided.  If  one  of  the  equal  parts  58, 
be  multiplied  by  the  number  of  parts,  8,  and  the  remainder  2  be 
added  to  the  product,  the  result  will  be  equal  to  466,  the  number 
divided. 

61.  DIVISION  is  the  operation  of  dividing  a  number  into 
two  equal  parts  ;  or,  of  finding  how  many  times  one  number 
contains  another. 

The  first  number,  or  number  by  which  we  divide,  is  called 
the  divisor. 

The  second  number,  or  number  to  be  divided,  is  called  the 
dividend. 

The  third  number,  or  result,  is  called  the  quotient 

The  quotient  shows  how  many  times  the  dividend  contains 
the  divisor. 

If  anything  is  left  after  division,  it  is  called  a  remainder. 

62.  There  are  three  parts  in  every  division,  and  sometimes 
four  :  1st,  the  dividend ;  2d,  the  divisor ;  3d,  the  quotient ; 
and  4th,  the  remainder. 

There  are  three  signs  used  to  denote  division  ;  they  are  the 
following : 

lS-f-4  expresses  that  18  is  to  be  divided  by  4. 
-^8        expresses  that  18  is  to  be  divided  by  4. 
4)18  expresses  that  18  is  to  be  divided  by  4. 
When  the  last  sign  is  used,  if  the  divisor  does  not  exceed 
12,  we  draw  a  line  beneath,  and  set  the  quotient  under  it.    If 
the  divisor  exceeds  12,  we  draw  a  curved  line  on  the  right  of 
the  dividend,  and  set  the  quotient  at  the  right. 

2.— When  the  seven  hundreds  are  divided  by  3,  what  is  the  unit  of 
the  quotient?  To  how  many  tens  is  the  undivided  hundred  equal? 
When  the  13  tens  arc  divided  by  8,  what  is  the  unit  of  the  quotient? 
Whun  the  9  uuits  arc  divided  by  #,  what  is  the  quotient  ? 

--How  is  the  division  of  the  remainder  expressed  ?  Read  the 
quotient.  If  there  be  a  remainder  after  division,  how  must  it  be  written  ? 

61.  What  is  division  ?    What  is  the  number  to  be  divided  called  ? 
What  is  the  number  called  by  which  we  divide?    What  is  the  answer 
called  ?     What  is  the  number  oalled  which  is  left  ? 

62.  Plow   many  parts  arc  there  in   division  ?    Name  them.     How 
many  signs  are  there  in  division  ?    Make  and  name  them  ? 


60  SHORT   DIVISION. 

SHORT  DIVISION. 

63.  SHORT  DIVISION  is  the  operation  of  dividing  when  the 
work  is  performed  mentally,  and  the  results  only  written 
down.  It  is  limited  to  the  cases  in  which  the  divisors  do  not 
exceed  12. 

Let  it  be  required  to  divide  30456  by  8. 

ANALYSIS — We  first  say,  8  in  3  we  cannot.  Then,  OPERATION. 

8  in  30,  3  times  and  6  over;  then  8  in  64,  8  times  ;  8)30456 
then  8  in  5,  0  times;  then,  8  in  50.  7  times:  hence, 

/  ooOT 

RULE  I. —  Write  the  divisor  on  the  left  of  the  dividend. 
Beginning  at  the  left,  divide  each  figure  of  the  dividend  by 
the  divisor,  and  set  each  quotient  figure  under  its  dividend 

II. — If  there  is  a  remainder,  after  any  division,  annex  (o  it 
the  next  figure  of  the  dividend,  and  divide  as  hcfnrp  „,  ^ 

III.  Jf  any  dividend  is  less  than  the  divisor,  write  0/br  the 
quotient  figure  and  annex  the  next  figure  of  the  dividend,  for 
a  new  dividend. 

IV.  If  there  is  a  remainder,  after  dividing  the  last  figure, 
set  the  divisor  under  it,  and  annex  the  result  to  the  quotient. 

PROOF. — Multiply  the  divisor  by  the  quotient,  and  to  the 
product  add  the  remainder,  when  there  is  one  ;  if  the  work 
is  right  the  result  will  be  equal  to  the  dividend. 

/ 

EXAMPLES. 

(1.)      (2.)       (3,)        (4) 
3)9369    4)73684    5)673420    6)825467 


Ans.    3123     18421     134684     137577f 

3        4 5 6_ 

Proof   9369     73684     673420     825467" 


5.  Divide  86434  by  2. 

6.  Divide  416710  by  4. 
7  Divide  641 40  by  5. 

8.  Divide  278943  by  6. 

9.  Divide  95040522  by  6. 

10.  Divide  75890496  by  8. 

11.  Divide  6794108  by  3. 

12.  Divide  21090431  by  9. 


13.  Divide  2345678964  by  6 
14  Divide  570196382  by  12 

15.  Divide  67897634  by  9. 

16.  Divide  75436298  by  12. 

17.  Divide  674189904  by  9. 

18.  Divide  1404967214 by  11. 

19.  Divide  27478041  by  10 
20  Divide  167484329  by  12. 


EQUAL  PARTS.  61 

21.  A  man  sold  his  farm  for  6756  dollars,  and  divided  the 
amount  equally  between  his  wife  and  5  children  :  how  much 
did  each  receive  ? 

22.  There  are  576  persons  in  a  train  of  12  cars :    how 
many  are  there  in  each  car  ? 

23.  If  a  township  of  land  containing  2304  acres  be  equally 
divided  among  8  persons,  how  many  acres  will  each  have  ? 

24.  If  it  takes  5  bushels  of  wheat  to  make  a  barral  of  flour, 
how  many  barrels  can  be  made  from  65890  bushels  ? 

25.  Twelve  things  make  a  dozen  :  how  many  dozens  are 
therein  2167284? 

26.  Eleven  persons  are  all  of  the  same  age,  and  the  sum 
of  their  ages  is  968  years  :  what  is  the  age  of  each  ? 

27.  How  many  barrels  of  flour  at  7  dollars  a  barrel  can  be 
bought  for  609463  dollars  ? 

28.  An  estate  worth  2943  dollars,  is  to  be  divided  equally 
among  a  father,  mother,  3  daughters  and  4  sons :  what  is 
the  portion  of  each  ? 

29.  A  county  contains  207360  acres  of  land  lying  in  9  town- 
ships of  equal  extent :  how  many  acres  in  a  township  ? 

30.  If  11  cities  contain  an  equal  number  of  inhabitants, 
and  the  whole  number  is  equal  to  3800247  :  how  many  will 
there  be  in  each  ? 

EQUAL   PARTS    OF    NUMBERS. 

64.  1.  If  any  number  or  thing  be  divided  into  two  equal 
parts,  one  of  the  parts  is  called  one-half:  one  half  of  a  single 
thing  is  written  thus  ;  J. 

2.  If  any  number  is  divided  into  three  equal  parts,  one  of 
the  parts  is  called  one-third,  which  is  written  thus  ;  \  ;  two 
of  the  parts  are  called  two-thirds:  which  are  written  thus  ;  f . 

3.  If  any  number  is  divided  into  four  equal  parts,  one  of 
the  parts  is  called  one-fourth,  which  is  written  thus  ;  J  ;  two 
of  the  parts  are  called  two-fourths,  and  are  written  thus  ;  £  ; 
three  of  them  are  called  three-fourths,  and  written  J  ;  and 
similar  names  are  given  to  the  equal  parts  into  which  any 
number  may  be  divided. 

63.  What  is  short  division  ?    How  is  it  generally  performed  ?    Give 
the  rule  ?    How  do  you  prove  short  division  ? 


62  EQUAL  PARTS 

4.  If  a  number  is  divided  into  five  equal  parts,  what  is  one 
of  the  parts  called  ?    Two  of  them  ?     Three  of  them  ?     Pour 
of  them  ? 

5.  If  a  number  is  divided  into  7  equal  parts,  what  is  one 
of  the  parts  called  ?     What  is  one  of  the  parts  called  when 
it  is  divided  into  8  equal  parts  ?     When  it  is  divided  into  9 
equal  parts  ?    When  it  is  divided  into  10  ?    When  it  is  divided 
into  11  ?     When  it  is  divided  into  12  ?  - 

6.  What  is  one-half  of  2?  of4?  of6?  ofS?  of  10?  of  12? 
of  14?  of  16?  of  18? 

7.  What  is  two-thirds  of  3  ? 

ANALYSTS —Two-thirds  of  three  are  two  times  one  third  of 
three.  ODe-third  of  three  is  1  ,  therefore,  two-thirds  of  three  are 
two  times  1,  or  2. 

Let  every  question  be  analyzed  in  the  same  manner. 

What  is  one-third  of  6  ?  2  thirds  of  6  ?  One-third  of  9  ? 
2  thirds  of  9  ?  One-third  of  12  ?  two-thirds  of  12  ? 

8.  What  is  one-fourth  of  4  ?  2  fourths  of  4  ?  3  fourths  of  4  ? 
What  is  one-fourth  of  8  ?  2  fourths  of  8  ?  3  fourths  of  8  ?  What 
is  one-fourth  of  12  ?  2  fourths  of  12  ?  3  fourths  of  12  ?     One- 
fourth  of  16  ?  2  fourths  of  16  ?  3  fourths  ? 

9.  What  is  one-seventh  of  7  ?    What  is  2  sevenths  of  7  ?  5 
sevenths?  6  sevenths?    What  is  one-seventh  of  14?  3  sev- 
enths ?  5  sevenths  ?  6  sevenths  ?    What  is  one-seventh  of  21  ? 
of  28  ?  of  35  ? 

10.  What  is  one-eighth  of  8?  of  16?  of  24?  of  32?  of 
40?  of  56? 

1 1 .  What  is  one-ninth  of  9  ?  2  ninths  ?  7  ninths  ?  6  ninths  ? 
5  ninths?  4  ninths?     What  is  one-ninth  of  18?  of  27?  of 
54?  of  72?  of  90?  of  108? 

12.  How  many  halves  of  1  are  there  in  2  ? 

ANALYSIS  — There  are  twice  as  many  halves  in  2  as  there  are 
in  1.  There  are  two  halves  in  1 ;  therefore,  there  are  2  times  2 
''halves  in  2,  or  4  halves. 

13  How  many  halves  of  1  are  there  in  3  ?    In  4  ?    In  5  ? 
In  6?  In  8?  In  10?  In  12? 

14  How  many  thirds  are  there  in  1  ?     How  many  thirds 
of  1  in  2?  In  3?  In  4?  In  5?  In  6?  In  9?  In  12? 

15.  How  many  fourths  are  there  in  1  ?  How  many  fourths 
of  1  in  2?  In  4?  In  6?  In  10?  In  12? 


OF   NUMBERS. 

16.  How  many  fifths  are  there  in  1  ?     How  many  fifths  of 

1  are  there  in  2  ?  In  3  ?  In  6  ?  In  1  ?  In  11  ?  In  12  ? 

17.  How  many  sixths  are  there  in  2  and  one-sixth  ?     In  3 
and  4  sixths  ?  In  5  and  2  sixths  ?  In  8  and  5  sixths  ? 

18.  How  many  sevenths  of  1  are  there  in  2  ?     In  4  and  3 
sevenths  how  many  ?     How  many  in  5  and  5  sevenths  ?     In  fc 
5  and  6  sevenths  ? 

19.  How  many  eighths  of  1  are  there  in  2  ?     How  many 
in  2  and  3  eighths  ?  In  2  and  5  eighths  ?  In  2  and  7  eighths? 
In  3  ?    In  3  and  4  eighths  ?  In  9  ?    In  9  and  5  eighths  ?    In 

10  ?  In  10  and  7  eighths  ? 

20.  How  many  twelfths  of  1  are  there  in  2  ?     In  2  and  4 
twelfths  how  many  ?     How  many  in  4  and  9  twelfths  ?     How 
many  in  5  and  10  twelfths?  In  6  and  9  twelfths?  In  10  and 

11  twelfths? 

21.  What  is  the  product  of  12  multiplied  by  3  and  one 
half,  (which  is  written  3J)  ? 

ANALYSIS. — Twelve  is  to  be  taken  3  and  one-half  times  (Art 
45).  Twelve  taken  £  times  is  6  ;  and  12  taken  three  times  is  36 ; 
therefore,  12  taken  ty  times  is  42. 

22.  What  is  the  product  of  10  multiplied  by  5J  ? 

23.  What  is  the  product  of  12  multiplied  by  3J  ? 

24.  What  is  the  product  of  8  multiplied  by  4  J  ? 

25.  What  will  9  barrels  of  sugar  cost  at  2§  dollars  a 
barrel? 

ANALYSIS. — Nine  barrels  of  sugar  will  cost  nine  times  as 
much  as  1  barrel.  If  one  barrel  of  sugar  costs  2f  dollars,  9 
barrels  will  cost  9  times  2f  dollars,  which  are  24  dollars.  For, 

2  thirds  taken  9  times  gives  18  thirds,  which  are  equal  to  6  ;  then 
9  times  2  are  18,  and  6  added  gives  24  dollars. 

26.  What  will  6  yards  of  cloth  cost  at  5§  dollars  a  yard  ? 

27.  What  will  12  sheep  cost  at.4J  dollars  apiece  ? 

28.  What  will  10  yards  of  calico  cost  at  9f  cents  a  yard  ? 

29.  What  will  8  yards  of  broadcloth  cost  at  7-J  dollars 
a  yard  ?  /  - 

30.  What  will  9  tons  of  hay  cost  at  9^  dollars  a  ton  ? 

31.  How  many  times  is  2J  contained  in  10  ? 

ANALYSIS. — Two  and  one-half  is  equal  to  5  halves ;  and  10  is 
equal  to  20  halves ;  then  5  halves  is  contained  in  20  halves  4 
times:  hence. 


LONG   DIVISION. 

In  all  similar  questions  change  the  divisor  and  dividend 
to  the  same  fractional  unit.  (Art.  144). 

32.  How  many  yards  of  cloth,  at  3J  dollars  a  yard,  can 
you  buy  for  14  dollars  ?  how  many  for  21  dollars  ? 

33.  If  oranges  are  3|  cents  apiece,  how  many  can  you  buy 
for  20  cents  ?       • '    ,: 

34.  If  1  yard  of  nbbon  costs  2f   cents,  how  many  yards 
can  you  buy  for  12  cents  ? 

35.  If  1  yard  'of  broadcloth  costs  3|  dollars,  how  many- 
yards  can  be  bought  for  33  dollars  ? 

36.  If  1  pound  of  sugar  costs  4J  cents,  how  many  pounds 
can  be  bought  for  36  cents  ?      / 

37.  How  many  times  is  5J  contained  in  44  ? 

38.  How  many  times  is  2|  contained  in  24  ? 

39.  How  many  lemons,  at  2|  cents  apiece,  can  you  buy 
for  32  cents  ? 

40.  How  many  yards  of  ribbon,  at  1^  cents  a  yard,  can 
you  buy  for  12  cents  ? 

LONG  DIVISION. 

65.  LONG  DIVISION  is  the  operation  of  finding  the  quotient 
of  one  number  divided  by  another,  and  embraces  the  case  of 
Short  Division,  treated  in  Art.  63. 

1.  Let  it  be  required  to  divide  7059  by  13. 

ANALYSIS. — The    divisor,    13,    is    not  OPERATION. 

contained    in     7     thousands ;     therefore,  .  „•              ^ 

there  are  no  thousands  in  the  quotient.  &  ^  «J  J&    '  §  m  -3 

We  then    consider  the  0  to  be  annex-  J2  s  g  '3      a  g  *3 

ed  to  the  7,  making  70  hundreds,  and  EH  W  EH  P    W  EH  P 

call  this  a  partial  dividend.  13)70  5  9(5  43 

The    divisor,    13,    is    contained    in    70  65 

hundreds,   5   hundreds    times    and  some-  — ^-£- 
thing    over.     To    find    how  much    over, 

multiply  13  by  5  hundreds  and  subtract  5  2 

the   product    65  from  70,  and   there  will  r            3  g 

remain     5     hundreds,     to    which    bring  «  q 

down   the  5   tens    and    consider    the  55  _r__ 
tens  a  new  partial  dividend. 

65.  What  is  long  division  ?    Does  it  embrace  the  case  of  short  divi- 
sion ?    What  is  u  partial  dividend  ? 


SIMPLE   NUMBERS.  65 

Then,  13  is  contained  in  55  tens,  4  tens  times  and  something 
over.  Multiply  13  by  4  tens  and  subtract  the  product,  52,  from 
55,  and  to  the  remainder  3  tens  bring  down  the  9  units,  and  con- 
sider the  39  units  a  new  partial  dividend. 

Then,  13  is  contained  in  39,  3  times.  Multiply  13  by  3,  and 
subtract  the  product  39  from  39,  and  we  find  that  nothing  remains. 

66.  PROOF. — Each  product  that  has  arisen  from  multiply- 
ing the  divisor  by  a  figure  of  the  quotient,  is  a  partial  product, 
and  the  sum  of  these  products  is  the  product  of  the  divisor 
and  quotient  (Art.  51,  XOTE).     Each  product  has  been  taken, 
separately,  from  the  dividend,  and  nothing  remains.     But, 
taking  each  product  away  in  succession,  leaves  the  same  re- 
mainder as  would  be  left  if  their  sum  were  taken  away  at 
once.     Hence,   the   number   543,   when  multiplied  by  the 
divisor,  gives  a  product  equal  to  the  dividend :  therefore,  543 
is  the  quotient  (Art.  61)  :  hence,  to  prove  division, 

Multiply  the  divisor  by  the  quotient  and  add  in  the  remain- 
der, if  any.  If  the  work  is  right,  the  result  will  be  the  same 
as  the  dividend. 

67.  Let  it  be  required  to  divide  2756  by  26. 

We  first  say,  26  in  27  once,  and  place  1  in  OPERATION. 

the  quotient.    Multiplying  by  1,  subtracting,  26)2756(106 

and  bringing  down  the  5,  we  have  15  for  the  26 

first  partial  dividend.     We  then  say,  26  in  15,  "^ 

0  times,  and  place  the  0  in  the  quotient.     We  156 

then  bring  down  the  6,  and  find  that  the  divisor  156 
is  contained  in  156,  6  times. 

If  anyone  of  the  partial  dividends  is  less  than  the  divisor,  write 
0  for  the  quotient  figure,  and  then  bring  down  the  next  figure, 
forming  a  new  partial  dividend. 

Hence,  for  Long  Division,  we  have  the  following 
KULE. — I.   Write  the  divisor  on  the  left  of  the  dividend. 

II.  Note  the  fewest  figures  of  the  dividend,  at  the  left, 
that  will  contain  the  divisor,  and  set  the  quotient  figure  at 
the  right. 


66.  What  is  a  partial  product  ?    What  is  the  sum  of  all  the  partial 
products  equal  to  ?     How  do  you  prove  division  ? 

67.  What  do  you  do  if  any  partial  dividend  is  less  than  the  divisor  ? 
What  is  the  rule  for  long  division  ? 


66 


LONG   DIVISION. 


III.  Multiply  the  divisor  by  the  quotient  figure,  subtract 
the  product  from  the  first  partial  dividend,  and  to  the  re- 
mainder annex  the  next  figure  of  the  dividend,  forming  a 
second  partial  dividend. 

TV.  find  in  the  same  manner  the  second  and  succeeding 
figures  of  the  quotient,  till  all  the  figures  of  the  dividend 
are  brought  down. 

NOTE  1. — There  arc  five  operations  in  Long  Division.  1st.  To 
write  down  the  numbers :  2d.  Divide,  or  find  how  many  times : 
3d.  Multiply :  4th.  Subtract :  5th.  Bring  down,  to  form  the  partial 
uividends. 

2.  The  product  of  a  quotient  figure  by  the  divisor  must  never 
be  larger  than  the  corresponding  partial  dividend :   if  it  is,  the 
quotient  figure  is  too  large  and  must  be  diminished. 

3.  When  any  one  of  the  remainders  is  greater  than  the  divisor, 
the  quotient  figure  is  too  small  and  must  be  increased. 

4.  The  unit  of  any  quotient  figure  is  the  same  as  that  of  the 
partial  dividend  from  which  it  is  obtained.      The  pupil  should 
always  name  the  unit  of  every  quotient  figure. 


EXAMPLES. 


1.  Divide  7574  by  54. 

OPERATION. 

54)7574/140 

54 


2.  Divide  67289  by  261. 

OPERATION. 

261)67289(257 
522 

1508 
1305 


2039 
1827 
212  Remainder, 


PROOF. 

140  Quotient. 
54  Divisor. 


560 
700 
7560 

14  Remainder. 
7574  Dividend. 


PROOF. 

261  Divisor. 
257  Quotient. 

1827 
1305 
522 

212  Remainder. 
-#7289  Dividend. 


SIMPLE  NUMBERS.                                  67 
3.  Divide  119836687  by  39407. 

OPERATION.  PROOF. 

39407)119836687(3041  39407  Divisor. 

118221  3041  Quotient. 

161568  39407 

157628  157628 

39407 .  118221 

39407  119836687  Dividend. 


4.  Divide  7210473  by  37. 

5.  Divide  147735  by  45. 

6.  Divide  937387  by  54. 

7.  Divide  145260  by  108 

8.  Divide  79165238  by  238. 


9.  Divide  62015735  by  78. 

10.  Divide  14420946  by  74. 

11.  Divide  295470  by  90. 

12.  Divide  1874774  by  162. 

13.  Divide  435780  by  216. 


14.  Divide  203812983  by  5049. 

15.  Divide  20195411808  by  3012. 

16.  Divide  74855092410  by  949998. 

17.  Divide  47254149  by  4674. 

18.  Divide  119184669  by  38473. 

19.  Divide  280208122081  by  912314. 

20.  Divide  293839455936  by  8405. 

21.  Divide  4637064283  by  57606. 

22.  Divide  352107193214  by  210472. 

23.  Divide  558001172606176724  by  2708630425. 

24.  Divide  1714347149347  by  57143. 

25.  Divide  6754371495671594  by  678957 

26.  Divide  71900715708  by  37149.     1 

27.  Divide  571943007145  by  37149. 

28.  Divide  671493471549375  by  47143. 

29.  Divide  571943007645  by  37149. 

30.  Divide  171493715947143  by  57007. 

31.  Divide  121932631112635269  by  987654321. 

NOTES. — 1.  How  many  operations  are  there  in  long  division  ?    Name 
them. 

2.  If  a  partial  product  is  greater  than  the  partial  dividend,  what  does 
it  indicate  ?     What  do  you  do  ? 

3.  What  do  you  do  when  any  one  of  the  remainders  is  greater  than 
the  divisor  ? 

4.  What  is  the  unit  of  any  figure  of  the  quotient  ?    When  the  divisor 
is  contained  in  simple  units,  what  will  be  the  unit  of  the  quotient  figure  ? 
When  it  is  contained  in  tens,  what  will  be  the  unit  of  the  quotient 
figure  ?    When  it  is  contained  in  hundreds  ?    In  thousands  ? 


68  LONG    DIVISION. 

08.    PRINCIPLES    RESULTING    FROM    DIVISION. 

NOTES. — 1st.  When  the  divisor  is  1,  the  quotient  will  be  equal 
to  the  dividend. 

„     2d.  When  the   divisor  is  equal  to  the  dividend,  the  quotient 
'  will  be  1. 

3d.  "When  the  divisor  is  less  than  the  dividend,  the  quotient 
will  be  greater  than  1.  The  quotient  will  be  as  many  times 
greater  than  1,  as  the  dividend  is  times  greater  than  the  divisor. 

4th.  When  the  divisor  is  greater  than  the  dividend,  the  quotient 
will  be  less  than  1.  The  qaot'ent  will  be  such  a  part  of  1,  as 
the  dividend  is  of  the  divisor. 


PROOF    OF    MULTIPLICATION. 

69.  Division  is  the  reverse  of  multiplication,  and  they 
prove  each  other.  The  dividend,  in  division,  corresponds  to 
the  product  in  multiplication,  and  the  divisor  and  quotient  to 
the  multiplicand  and  multiplier,  Avhich  are  factors  of  the  pro- 
duct :  hence, 

If  the  product  of  two  numbers  be  divided  by  the  multipli- 
cand, the  quotient  will  be  the  multiplier  ;  or,  if  it  be  divided 
by  the  multiplier,  the  quotient  will  be  the  multiplicand. 

EXAMPLES. 

3679  Multiplicand  3679J1203033(327 

327  -Multiplier.  11037 


25753  9933 

7358  7358 


11037  25753 

1203033  Product.  25753 

2.  The     multiplicand  is     61835720,     and     the    product 
8162315040  :  what  is  the  multiplier  ? 

3.  The   multiplier   is   270000  ;    now  if    the   product   be 
1315170000000,  what  will  be  the  multiplicand? 

4.  The  product  is  68959488,  the  multiplier  96 :  what  is 
the  multiplicand  ? 

5.  The   multiplier   is    1440,   the   product    10264849920  : 
what  is  the  multiplicand  ? 

6.  The    product    is    6242102428164,    the    multiplicand 
6795634  :  what  is  the  multiplier  ? 


CONTRACTIONS   IN  MULTIPLICATION.  G9 

CONTRACTIONS  IN  MULTIPLICATION. 

70.   To  multiply  by  25. 
1.  Multiply  275  by  25.' 

ANALYSIS. — If  we  annex  two    ciphers  to  the  mul-  OPERATION-. 

tiplicand,    we    multiply  it    by   100   (Art.   55):     this  4)27500 

product  is  4  times  too  great ;   for  the  multiplier  is  •  /»,,7- 
but  one-fourth  of  100  ;  hence,  to  multiply  by  25, 

Annex  two  ciphers  to  the  multiplicand  and  divide  the 
result  by  4. 


EXAMPLES. 


1.  Multiply  127  by  25. 

2.  Multiply  4269  by  25. 


3.  Multiply  87504  by  25. 

4.  Multiply  7-04963  by  25. 


71.   To  multiply  by  12  J 
1.  Multiply  326  by  m. 

ANALYSIS. — Since   12^   is    one-eighth  of    100,  OPERATION. 

Annex  two  ciphers  to  the  multiplicand  and  di-  8)32600 

vide  the  result  by  8.  4.075 


EXAMPLES. 


1.  Multiply  284  by  12J. 

2.  Multiply  376  by  121. 


3.  Multiply  4740  by  12£. 

4.  Multiply  70424  by  12 


72.   To  multiply  by  33* 
1.  Multiply  675  by  33J. 

ANALYSIS. — Annexing  two  ciphers  to  the  mul-  OPERATION. 
tiplicand,  multiplies  it  by  100:  but  the  multiplier  3)67500 
is  but  one-third  of  100 :  hence, 

Annex  two  ciphers  and  divide  the  result  ly  3. 


EXAMPLES. 


1.  Multiply  889626  by  33J. 
2    Multiply  740362  by  33J. 


3.  Multiply  5337756  by  33J. 

4.  Multiply  2221086  by  33i. 


68.  When  the  divisor  is  1,  what  is  the  quotient?  Wheii  the  divisor 
is  equal  to  the  dividend,  what  is  the  quotient  ?  When  the  divisor  is  less 
than  the  dividend,  how  does  the  quotient  compare  with  1  ?  When  the  di- 
visor is  greater  than  the  dividend,  how  doas  the  quotient  compare  with  1  ? 

09.  If  a  product  be  divided  by  one  of  the  factors,  what  is  the  quotient  ? 


70 


CONTRACTIONS   IN   MULTIPLICATION. 


73.   To  multiply  by  125. 
1.  Multiply  375  by  125. 

ANALYSIS. — Annexing  three  ciphers  to  the  mul- 
tiplicand, multiplies  it  by  1000  :  but  125  is  but 
one-eighth  of  one  thousand  :  hence, 

Annex  three  ciphers  and  divide  the  result  by  8. 


OPERATION. 

8)375000 

46875 


EXAMPLES. 


1.  Multiply  29632  by  125. 

2.  Multiply  8796704  by  125. 


3.  Multiply  970406  by  125. 

4.  Multiply  704294  by  125. 


74.  By  reversing  the  last  four  processes,  we  have  the  four 
folio  whig  rules : 

1.  To  divide  any  number  by  25  ; 

Multiply  the  number  by  4,  and  divide  the  product  by  100. 

2.  To  divide  any  number  by  12£. 

Multiply  the  number  by  8,  and  divide  the  product  by  100. 

3.  To  divide  any  number  by  33 \  : 

Multiply  the  number  by  3,  and  divide  the  product  by  100. 

4.  To  divide  any  number  by  125  : 

Multiply  by  8,  and  divide  the  product  by  1000. 

EXAMPLES. 


1. 

2. 
3. 
4. 
6. 

6. 

7. 
8. 

Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 

3175  by  25. 
106725  by  25. 
2187600  by  25. 
2426225  by  25. 
1762405  by  25. 
4075  by  12J. 
3550  bv  12J. 
59262$  by  12J. 

9. 
10. 
11. 
12. 
,13. 
14 
15. 
16. 

Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 

880300  by  12i. 
22500  by  33J. 
654200  by  33J. 
7925200  by  33£. 
4036200  by  33f  . 
93750  by  125. 
3007875  by  125. 
6758625  by  125. 

70.  What  is  the  rule  for  multiplying  by  25  ? 

71.  What  is  the  rule  for  multiplying  by  12*  ? 

72.  What  is  the  rule  for  multiplying  by  88*  ? 

73.  What  is  the  rule  for  multiplying  by  135? 


CONTRACTIONS   IN  DIVISION.  71 


CONTRACTIONS  IN  DIVISION. 

75.  Contractions  in  Division  are  short  methods  of  finding 
the  quotient,  when  the  divisors  are  composite  numbers. 

CASE    I. 

76.   When  the  divisor  is  a  composite  number. 

1.  Let  it  be  required  to  divide  1407  dollars  equally  among 
2i  rnen.  Here  the  factors  of  the  divisor  are  7  and  3. 

ANALYSIS.— Let    the    1407  dollars 

be  first   divided  into   7  equal   piles.  OPERATION. 

Each   pile  will   contain   201    dollars.  7)1407 

Let  each  pile  be  now  divided  into  3  0,  .... .    ,   ,          , . 

equal  park     Each  part  will  contain  S)201    lst  quotient. 

67  dollars,  and  the  number  of  parts  G7  quotient  sought, 
will  bo  21  :  hence  the  following 

RULE. — Divide  the  dividend  by  one  of  the  factors  of  the 
divisor  ;  tlien  divide  the  quotient,  thus  arising,  by  a  second 
factor,  and  so  on,  till  every  factor  has  been  used  as  a  divisor : 
the  last  Quotient  will  be  the  answer. 

EXAMPLES. 

Divide  the  following  nnmbers  by  the  factors  ; 


1.  1260  by  12  —  3x4. 

2.  18576  by  48=4  x  12. 

3.  9576  by  72  =  9x8. 

4.  19296  by  %=12x8. 


5.  55728  by4x  9x4=14 4. 

6.  92880  by  2x2x3x2x2. 

7.  57888  by4x2x2x2. 

8.  154368  by  3  x  2  x  fc. 


NOTE. — It  often  happens  that  there  are  remainders  after  some 
of  the  divisions      How  are  we  to  find  ihe  true  remainder? 


74. — 1.  What  is  the  rule  for  divicling  by  25  ? 

2.  What  is  the  rule  for  dividing  by  12*  ? 

3.  What  is  the  rule  for  dividing  by  33*  ? 

4.  What  is  the  rale  for  dividing  by  125  ? 

75.  What  are  contractions  in  division  ?    What  is  a  composite  num- 
ber? 

76.  What  is  the  rule  for  division    when  the  divisor  is  a  composite 
number  ? 


72  CONTRACTIONS. 

77.  Let  it  be  required  to  divide  751  grapes  into  16  equal 
parts. 

(4)751 

4  x  4  =  16  -j  4)18T  ....  3  first  remainder. 
40  ....  3x4  =  12 
3 


15  true  rem.    4ns.  -4S}|. 
NOTE. — The  factors  of  the  divisor  16,  are  4  and  4. 

ANALYSIS.— If  751  grapes  be  divided  by  4,  there  will  be  187 
bunches,  each  containing  4  grapes,  and  8  grapes  over.  The  unit 
of  187  is  one  bunch  ;  that  is,  a  unit  4  times  <(s  great  as  1  grape. 

If  we  divide  187  bunches  by  4,  we  shall  have  46  piles,  each 
containing  4  bunches,  and  3  bunches  over :  here,  again,  the  unit 
of  the  quotient  is  4  times  as  great  as  the  unit  of  the  dividend. 

If,  now  we  wish  to  find  the  number  of  grapes  not  included  in 
the  46  piles,  we  have  3  bunches  with  4  grapes  in  a  bunch,  and 
3  grapes  besides :  hence,  4  x  3  =  12  grapes ;  and  adding  3 
grapes,  we  have  a  remainder,  15  grapes ;  therefore,  to  find  the 
remainder,  in  units  of  the  given  dividend : 

I.  Multiply  the  last  remainder  by  the  last  divisor  but  onr, 
and  add  in  the  preceding  remainder : 

II.  Multiply  this  result  by  the  next  preceding  divisor, 
and  add  in  the  remainder,  and  so  on,  till  you  reach  the 
unit  of  the  dividend. 

EXAMPLES, 

1.  Let  it  be  required  to  divide  43720  by  45. 
3)43720 

5)14573  .  l  =  lstrem.     1x5  +  3-8; 
3)2914  .  3= 3d  rem.     8x3  +  1  =  25 

971  .  1  =  3d  rein.  25  true  r era. 

Divide  the  following  numbers  by  the  factors,  for  the  divisors  : 


2.  956789  by  7x8  =  56. 

3.  4870029  by  8x9  =  72. 

4.  674201  by*10x  11  =  110. 

5.  4-15767  by  12x12  =  144. 


6.  1913578  by  7x2x3  =  42. 

7.  146187  by  3x5x7  =  105. 

8.  26964  by  5x2  x  11  =  110. 

9.  93696  by  3x7x11  =  231. 


77.  Give  the  rule  for  the  remainder. 


IN   DIVISION.  73 

CASE    II. 

78.   When  the  Divisor  is  10,  100,  1000,  &c. 

ANALYSIS. — Since  any  number  is  made  up  of  units,  tens,  hun- 
dreds, &c.  (Art.  28),  the  number  of  tens  in  any  dividend  will 
denote  how  many  times  it  contains  1  ten,  and  the  units  "will  be  the 
remainder.  The  hundreds  will  denote  how  many  times  the  divi- 
dend contains  1  hundred,  and  the  tens  and  units  will  be  thi3  remain- 
der ;  and  similarly,  when  the  divisor  is  1000,  10000,  &c. ;  hence, 

RULE.—  Cut  off  from  the  right  hand  as  many  figures  as 
there  are  ciphers  in  the  divisor — the  figures  at  the  left  ivill  be 
the  quotient,  and  those  at  the  right,  the  remainder. 


EXAMPLES. 


1.  Divide  49763  by  10. 

2.  Divide  7641200  by  100. 


3.  Divide  496321  by  1000. 

4.  Divide  6i9T8  by  10000. 


CASE    III. 

79.   When  there  are  ciphers  on  the  right  of  the  divisor. 

I.  Let  it  be  required  to  divide  67389  by  700. 

ANALYSIS. — We  may  regard  the  OPERATION. 

divisor  as  a  composite   number,  of     7|00)673[89 
which  the   factors  are   7  and   100. 
We  first  divide  by  100  by  striking 

off  the  89,  and  then  find  that  7  is  189  true  remain, 

contained  in  the  remaining  figures,  "  ^ns    96—-— 

90  times,  with  a  remainder  of  1  ; 

this  remainder  we  multiply  by  100,  and  then  add  89,  forming  the 
true  remainder  189  :  to  the  quotient  96,  we  annex  189  divided  by 
700,  for  the  entire  quotient :  hence,  the  following 

RULE  I. — Cut  off"  the  ciphers  by  a  line,  and  cut  off"  the 
same  number  of  figures  from  the  right  of  the  dividend. 

II.  Divide  the  remaining;  figures  of  the  dividend  by  the 
remaining  figures  of  the  divisor,  and  annex  to  the  remainder, 
if  there  be  one,  the  figures  cut  off  from  the  dividend :  this  will 
form  the  true  remainder 

EXAMPLES. 
1.  Divide  8749632  by  37000. 

78.  How  do  you  divide  when  the  divisor  is  1  with  ciphers  annexed? 
Give  the  reason  of  the  rule. 

79.  How  do  you  divide  when  there  are  ciphers  on  the  right  of  the 
divisor  ?    How  do  you  form  the  true  remainder  ? 


APPLICATIONS. 

371000)87491632(236 
74 


Ans.  236JJJJJ. 


17 
Divide  the  following  numbers  : 

2.  986327  by  210000. 

3.  876000  by  6000. 

4.  36599503  by  400700. 


5.  5714364900  by  36500. 

6.  18490700  by  73000. 

7.  70807149  by  31500. 


APPLICATIONS. 

80.  Abstractly,  the  object  of  division  is  to  find  from  two 
given  numbers  a  third,  which,  multiplied  by  the  first,  will 
produce  the  second.  Practically,  it  has  three  objects  : 

1.  Knowing  the  number  of  things  and  their  entire  cost,  to 
find  the  price  of  a  single  thing  : 

2.  Knowing  the  entire  cost  of  a  number  of  things  and  the 
price  of  a  single  thing,  to  find  the  number  of  things  : 

3.  To  divide  any  number  of  things  into  a  given  number  of 
equal  parts. 

For  these  cases,  we  have  from  the  previous  principles 
(page  57),  the  following 

RULES. 

I.  Divide  the  entire  cost  by  the  number  of  the  things  : 
the  quotient  will  be  the  price  of  a  single  thing. 

II.  Divide  the  entire  cost  by  the  price  of  a  single  thing : 
the  quotient  will  be  the  number  of  things. 

III.  Divide  the  whole  number  of  things  by  the  number  of 
parts  into  which  they  are  to  be  divided :  the  quotient  will 
be  the  number  in  each  part. 

QUESTIONS    INVOLVING    THE    PREVIOUS    RULES. 

1.  Mr.  Jones  died,  leaving  an  estate  worth  4500  dollars,  to 
be  divided  equally  between  3  daughters  and  2  sons  :  what 
was  the  share  of  each  ? 

80.  What  is  the  object  of  division,  abstractly?  How  many  objects  has 
it,  practically  ?  Name  the  three  objects.  Give  the  rules  for  the  three  cases. 


APPLICATIONS.  75 

2.  What  number  must  be  multiplied  by  124  to  produce 
40796? 

3.  The  sum  of  19125  dollars  is  to  be  distributed  equally 
among  a  certain  number  of  men,  each  to  receive  425  dollars  : 
how  many  men  are  to  receive  the  money  ? 

4.  A  merchant  has  5100  pounds  of  tea,  and  wishes  to  pack 
it  in  60  chests  :  how  much  must  he  put  in  each  chest  ? 

5.  The  product  of  two  numbers  is  51679680,  and  one  of 
the  factors  is  615  :  what  is  the  other  factor  ? 

6.  Bought  156  barrels  of  flour  for  1092  dollars,  and  sold 
the  same  for  9  dollars  per  barrel :  how  much  did  I  gain  ? 

7.  Mr.  James  has  14  calves  worth  4  dollars  each,  40  sheep 
worth  3  dollars  each ;  he  gives  them  all  for  a  horse  worth 
150  dollars  :  what  does  he  make  or  lose  by  the  bargain  ? 

8.  Mr.  Wilson  sells  4  tons  of  hay  at  12  dollars  per  ton, 
80  bushels  of  wheat  at  1  dollar  per  bushel,  and  takes  in 
payment  a  horse  worth  65  dollars,  a  wagon  worth  40  dollars, 
and  the  rest  in  cash  :  how  much  money  did  he  receive  ? 

9.  How  many  pounds  of  coffee,  worth  12  cents  a  pound, 
must  be  given  for  368  pounds  of  sugar,  worth  9  cents  a 
pound  ? 

10.  The  distance  around  the  earth  is  computed  to  be  about 
25000  miles :  how  long  would  it  take  a  man  to  travel  that 
distance,  supposing  him  to  travel  at  the  rate  of  35  miles  a 
day? 

11.  If  600  barrels  of  flour  cost  4800  dollars,  what  will 
21 7 2  barrels  cost? 

12.  If  the  remainder  is  17,  the  quotient  610,  and  the  divi- 
dend 45767,  what  is  the  divisor? 

13.  The  salary  of  the  President  of  the  United  States  is 
25000  dollars  a  year :  how  much  can  he  spend  daily  and 
save  of  his  salary  4925  dollars  at  the  end  of  the  year  ? 

14.  A  farmer  purchased  a  farm  for  which  he  paid  18050 
dollars.     He  sold  50  acres  for  60  dollars  an  acre,  and  the  re- 
mainder stood  him  in  50  dollars  an  acre :  how  much  land 
did  he  purchase  ? 

15.  There   are   31173  verses  in   the   Bible:    how  many 
verses  must  be  read  each  day,  that  it  may  be  read  through 
in  a  year  ? 

16.  A  farmer  wishes  to  exchange  250  bushels  of  oats  at 
42  cents  a  bushel,  for  flour  at  7  dollars  per  barrel  :  how  many 
barrels  will  he  receive  ? 


76  APPLICATIONS. 

It.  The  owner  of  an  estate  sold  240  acres  of  land  and  had 
312  acres  left :  how  many  acres  had  he  at  first  ? 

18.  Mr.  James  bought  of  Mr.  Johnson  two  farms,  one  con- 
taining 250  acres,  for  which  he  paid  85  dollars  per  acre  ;  the 
second  containing  175  acres,  for  which  he  paid  70  dollars  an 
acre  ;  he  then  sold  them  both  for  75  dollars  an  acre  :  did  he 
make  or  lose,  and  how  much  ? 

19.  A  farmer  has  279  dollars  with  which  he  wishes  to  buy 
cows  at  25  dollars,  sheep  at  4  dollars,  and  pigs  at  2  dollars 
apiece,  of  each  an  equal  number  :  how  many  can  he  buy  of 
each  sort  ? 

20.  The  sum  of  two  numbers  is  3475,  and  the  smaller  is 
1162  :  what  is  the  greater  ? 

21.  The  difference  between  two  numbers  is  1475,  and  the 
greater  number  is  5760  :  what  is  the  smaller  ? 

22.  If  the  product  of  two  numbers  is  346712,  and  one  of 
the  factors  is  76  :  what  is  the  other  factor? 

23.  If  the  quotient  is  482,  and  the  dividend  135442  :  what 
is  the  divisor  ? 

24.  A  gentleman  bought  a  house  for  two  thousand  twenty- 
five  dollars,  and  furnished  it  for  seven  hundred  and  six  dol- 
lars ;  he  paid  at  one  time  one  thousand  and  ten  dollars,  and 
at  another  time  twelve  hundred  and  seven  dollars  :  how  much 
remained  unpaid  ? 

25.  At  a  certain  election  the  whole  number  of  votes  cast 
for  two  opposing  candidates  was  12672:  the  successful  can- 
didate received  316  majority  :  how  many  votes  did  each  re- 
ceive ? 

26.  Mr.  Place  purchased  15  cows :  he  sold  9  of  them  for 
35  dollars  apiece,  and  the  remainder  for  32  dollars  apiece, 
when  he  found  that  he  had  lost  123  dollars  :  how  much  did 
he  pay  apiece  for  the  cows  ? 

27.  Mr.  Gill,  a  drover,  purchased  36  head  of  cattle  at  64 
dollars  a  head,  and  88  sheep  at  5  dollars  a  head ;  he  sold  the 
cattle  at  one-quarter  advance  and  the  sheep  at  one-fifth  ad- 
vance :  how  much  did  he  receive  for  both  lots  ? 

28.  Mr.  Nelson  supplied  his  farm  with  4  yoke  of  oxen  at 
93  dollars  a  yoke  ;  4  plows  at  11  dollars  apiece  ;  8  horses  at 
97  dollars  each ;  and  agrees  to  pay  for  them  in  wheat  at 
1  dollar  and  a  half  per  bushel ;  how  many  bushels  must  he 
give  ? 


APPLICATIONS.  77 

29.  If  a  man's  salary  is  800  dollars  a  year  and  his  expenses 
425  dollars,  how  many  years  will  elapse  before  he  will  be 
worth  10000  dollars,  if  he  is  worth  2500  dollars  at  the  pre- 
sent time  ? 

30.  How  long  can  125  men  subsist  on  an  amount  of  food 
that  will  last  1  man  4500  days  ? 

31.  A  speculator  bought  512  barrels  of  flour  for  3584  dol- 
lars and  sold  the  same  for  4608  dollars :  how  much  did  he 
gain  per  barrel  ? 

32.  A  merchant  bought  a  hogshead  of  molasses  containing 
96  gallons  at  35  cents  per  gallon  ;  but  26  gallons  leaked  out, 
and  he  sold  the  remainder  at  50  cents  per  gallon  :  did  he 
gain  or  lose,  and  how  much  ? 

33.  Two  persons  counting  their  money,  together  they  had 
342  dollars  ;  but  one  had  28  dollars  more  than  the  other : 
how  many  had  each  ? 

34.  Mrs.  Louisa  Wilsie  has  3  houses  valued  at  12530  dol- 
lars, 11324  dollars,  and  9875  dollars :  also  a  farm  worth  6720 
dollars.     She  had  a  daughter  and  2  sons.     To  the  daughter 
she  gives  one-third  the  value  of  the  houses  and  one-fourth  the 
value  of  the  farm,  and  then  divides   the  remainder  equally 
among  the  boys  :  how  much  did  each  receive  ? 

35.  A  person  having  a  salary  of  1500  dollars,  saves  at  the 
end  of  the  year  405  dollars  :  what  were  his  average  daily 
expenses,  allowing  365  days  to  the  year  ? 

36.  Mr.    Bailey   has    7    calves   worth   4   dollars    apiece, 
9  sheep  worth  3  dollars  apiece,  and  a  fine  horse  worth  175 
dollars.     He  exchanges  them  for  a  yoke  of  oxen  worth  125 
dollars  and  a  colt  worth  65  dollars,  and  takes  the  balance  in 
hogs  at  8  dollars  apiece :  how  many  does  he  take  ? 

37.  Mr.  Snooks,  the  tailor,  bought  of  Mr.  Squire,  the  mer- 
chant, 4  pieces  of  cloth ;   the  first  and  second  pieces  each 
measured  45  yards,  the  third  47  yards,  and  the  fourth  53 
yards  ;  for  the  whole  he  paid  760  dollars  :  what  did  he  pay 
for  35  yards  ? 

38.  Mr.  Jones  has  a  farm  of  250  acres,  worth  125  dollars 
per  acre,  and  offers  to  exchange  with  Mr.  Gushing,  whose 
farm  contains  185  acres,  provided  Mr.  Gushing  will  pay  him 
20150   dollars   difference:    what   was   Mr.   Cushing's    farm 
valued  at  per  acre  ? 


78  APPLICATIONS. 

39.  The  volcano  in  the  island  of  Bourbon,  in  1796,  threw 
out  45000000  cubic  feet  of  lava  :  how  long  would  it  take  25 
carts  to  carry  it  off,  if  each  cart  carried  12  loads  a  day,  and 
40  cubic  feet  at  each  load  ? 

40.  The  income  of  the  Bishop  of  Durham,  in  England,  is 
292  dollars  a  day  ;  how  many  clergymen  would  this  support 
in  a  salary  of  730  dollars  per  annum  ? 

41.  The  diameter  of  the  earth  is  7912  miles,  and  the  diame- 
ter of  the  sun  112  times  as  great :  what  is  the  diameter  of  the 
sun? 

42.  By  the  census  of  1850,  the  whole  population  of  the 
United  States  was  23191876  ;  the  number  of  births  for  the 
previous  year  was  629444  and  the  number  of  deaths  324394  : 
supposing  the  births  to  be  the  only  source  of  increase,  what 
was  the  population  at  the  beginning  of  the  previous  year  ? 

43.  Mr.  Sparks  bought  a  third  part  of  neighbor  Spend- 
thrift's farm  for  2750  dollars.     Mr.  Spendthrift  then  sold  half 
the  remainder  at  an  advance  of  250  dollars,  and  then  Mr. 
Sparks  bought  what  was  left  at  a  further  advance  of  250 
dollars  :  how  much  money  did  Mr.  Sparks  pay  Mr.  Spend- 
thrift, and  what  did  he  get  for  his  whole  farm  ? 

44.  George  Wilson  bought  24  barrels  of  pork  at  14  dollars 
a  barrel ;  one-fourth  of  it  proved  damaged,  and  he  sold  it  at 
half  price,  and  the  remainder  he  sold  at  an  advance  of  3  dol- 
lars a  barrel :  did  he  make  or  lose  by  the  operation,  and  how 
much  ? 

45.  A  miller  bought  320  bushels  of  wheat  for  576  dollars, 
and  sold  256  bushels  for  480  dollars :  what  did  the  remain- 
der cost  him  per  bushel  ? 

46.  A  merchant  bought  117  yards  of  cloth  for  702  dollars, 
and  sold  76  yards  of  it  at  the  same  price  for  which  he  bought 
it ;  what  did  the  cloth  sold  amount  to  ? 

47.  If  46  acres  of  land  produce  2484  bushels  of  corn  ;  how 
many  bushels  will  1 20  acres  produce  ? 

48.  Mr.  J.  Williams  goes  into  business  with  a  capital  of 
25000  dollars  ;  in  the  first  year  he  gains  2000  ;  in  the  second 
year  3500  dollars  ;  in  the  third  year  4000  dollars  ;  he  then 
invests  the  whole  in  a  cargo  of  tea  and  doubles  his  money  ; 
he  then  took  out  his  original  capital  and  divided  the  residue 
equally    among    his  5  "children :  what  was  the  portion  of 
each  ? 


UNITED   STATES   MONEY.  79 


UNITED    STATES    MONEY. 

81.  Numbers  are  collections  of  units  of  the  same  kind. 
In  forming  these  collections,  we  first  collect  the  lowest  or  pri- 
mary units,    until   we   reach   a  certain  number ;    we   then 
change  the  unit  and  make  a  second  collection,  and  after 
reaching  a  certain  number  we  again  change  the  unit,  and  so  on. 

In  abstract  numbers,  we  first  collect  the  units  1  till  we 
reach  ten  ;  we  then  change  the  unit,  to  1  ten,  and  collect  till 
we  reach  10  ;  we  then  change  the  unit  to  100,  and  so  on. 

A  SCALE  expresses  the  relations  between  the  orders  of  units, 
in  any  number.  There  are  two  kinds  of  scales,  uniform  and 
varying.  In  the  abstract  numbers,  the  scale  is  uniform,  the 
units  of  the  scale  being  10,  at  every  step. 

82.  United  States  money  is  the  currency  established  by  Con- 
gress, A.D.  1786.    The  names  or  denominations  of  its  units  are, 
Double  Eagles,  Eagles,  Dollars,  Dimes,  Cents,  and  Mills. 

The  coins  of  the  United  States  are  of  gold,  silver,  and  cop- 
per, and  are  of  the  following  denominations  : 

1.  Gold :    Double-eagle,    eagle,   half-eagle,   three-dollars, 
quarter-eagle,  dollar. 

2.  Silver:  Dollar,  half-dollar,  quarter-dollar,  dime,  half- 
dime,  and  three-cent  piece. 

3.  Copper  :  Cent,  half-cent. 

TABLE. 


10  Mills    make  1  Cent,  Marked  ct. 

10  Cents     - 

-    1  Dime, 

-    -    d. 

10  Dimes    - 

-    1  Dollar, 

-    -    $. 

10  Dollars  - 

-    1  Eagle, 

-    -    E. 

Mills. 

Cents. 

Dimes. 

Dollars. 

Eagles. 

10 

=   1 

100 

=   10 

=   1 

1000 

=  100 

=   10 

=  1 

10000 

=   1000 

=   100 

=   10 

=  1 

81.  "What  are  numbers?  How  are  numbers  formed?  How  are  sim- 
ple numbers  formed  ?  What  is  the  scale  ?  What  is  the  primary  unit 
in  simple  numbers  ? 


80  UNITED   STATES   MONEY. 

83.  It  is  seen,  from  the  above  table,  that  in  United  States 
money,  the  primary  unit  is  1  mill  ;  the  units  of  the  scale,  in 
passing  from  mills  to  cents,  are  10.  The  second  unit  is  1 
cent,  and  the  units  of  the  scale,  in  passing  to  dimes,  are  10. 
The  third  unit  is  1  dime,  and  the  units  of  the  scale  in  passing 
to  dollars,  are  10.  The  fourth  unit  is  1  dollar,  and  the  units 
of  the  scale  in  passing  to  eagles,  are  10.  This  scale  is  the 
same  as  in  simple  numbers  ;  therefore, 

The  units  of  United  States  money  may  be  added,  sub- 
tracted, multiplied,  and  divided,  by  the  same  rules  that 
have  already  been  given  for  simple  numbers. 

NUMERATION  TABLE. 


5  7,  is  read  5  cents  and  7  mills,  or  57  mills. 
1  6  4,  -      -  16  cents  and  4  mills,  or  164  mills. 
6  2.  1  2  0,  -      -  62  dollars  12  cents  and  no  mills. 
27.623,-      -  27  dollars  62  cents  and  3  mills. 
4  0.  0  4  1,  -      -  40  dollars  4  cents  and  1  mill. 

The  period,  or  separatrix,  is  generally  used  to  separate  the 
cents  from  the  dollars.  Thus  $67.256  is  read  67  dollars  25 
cents  and  6  mills.  Cents  occupy  the  two  first  places  on  the 
right  of  the  period,  and  mills  the  third. 

United  States  money  is  read  in  dollars,  cents  and  mills. 

82.  What  is    United   States  money?     What  are  the  names  of   its 
units  ?     What  are  the    coins  of  the  United   States  ?     Which   gold  ? 
Which  silver  ?     Which  copper  ? 

83.  In  United  States  money  what  is  the  primary  unit?    What  is  the 
Hcale  in  passing  from  one  denomination  to  another?    I  low  does  this 
compare  with    the   scale   in   simple  numbers  ?     What  then  follows  V 
What  is  used  to  separate  dollars  from  cents  ?    How  is  United  States 
money  read  ? 

84.  What  is  reduction  ?    How  many  kinds  of  reduction  are  there  ? 
Name  them.     How  may  cents  be  changed  into  mills?    How  may  dol- 
lars be  changed  into  cents  ?    How  into  mills  ? 


UNITED   STATES   MONEY.  81 


REDUCTION  OF  UNITED  STATES  MONEY. 

84.  Reduction  of  United  States  Money  is  changing  the 
unit  from  one  denomination  to  that  of  another,  without  altering 
the  value  of  the  number.  It  is  divided  into  two  parts  : 

1st.  To  reduce  from  a  greater  unit  to  a  less,  as  from  dol- 
lars to  cents. 

2d.  To  reduce  from  a  less  unit  to  a  greater,  as  from  mills 
to  dollars. 

85.   To  reduce  from  a  greater  unit  to  a  less. 
From  the  table  it  appears, 

1st.  That  cents  may  be  changed  into  mills  by  annexing 
one  cipher. 

2d.  That  dollars  may  be  changed  into  cents  by  annexing 
two  ciphers,  and  into  mills  by  annexing  three  ciphers. 

3d.  That  eagles  may  be  changed  into  dollars  by  annexing 
one  cipher. 

The  reason  of  these  rules  is  evident,  since  10  mills  make  a 
cent,  100  cents  a  dollar,  and  1000  mills  a  dollar  and  10 
dollars  1  eagle. 

EXAMPLES. 

1.  Reduce  25  eagles,  14  dollars,  85  cents  and  6  mills  to 
the  denomination  of  mills. 

OPERATION. 

25  eagles =250  dollars, 
add     14  dollars, 

"264  dollars =2 64 00  cents, 
add        -  85  cents, 

26485  cents=264850  mills, 
add       -  -     6  mills, 

Ans.  264856  mills. 

2.  In  3  dollars  60  cents  and  5  mills,  how  many  mills  ? 
3  dollars =300  cents, 

60  cents, 

160  =  3600  mills,  to  which  add  the  5  mills. 
6 


82  REDUCTION   OF 

3.  In  37  dollars  31  cents  8  mills,  how  many  mills  ? 

4.  In  375  dollars  99  cents  9  mills,  how  many  mills  ? 

5.  How  many  mills  in  67  cents  ? 

6.  How  many  mills  in  $54  ? 

7.  How  many  cents  in  $125  ? 

8.  In  $400,  how  many  cents  ?     How  many  mills  ? 

9.  In  $375,  how  many  cents  ?     How  many  mills  ? 

10.  How  many  mills  in  $4  ?  In  $6  ?  In  $10.14  cents. 

11.  How  many  mills  in  $40.36  cents  8  mills  ? 

12.  How  many  mills  in  $71.45  cents  3  mills  ? 

86.  To  reduce  from  a  less  unit  to  a  greater. 
1.  How  many  dollars,  cents  and  mills  in  26417  mills? 

ANALYSIS. — We  first  divide  the  mills  by  10,  OPERATION. 

giving  2641  cents  and  7  mills  over;  we  then  10)264117 

divide  the  cents  by  100,  giving  26  dollars,  and  100)26141 

41  cents  over :  hence  the  answer  is  26  dollars  *!>„  .  -.  *, 
41  cents  and  7  mills :  therefore, 

I.  To  reduce  mills  to  cents  :  cut  off  the  right  hand  figure. 

II.  To  reduce  cents  to  dollars  :  cut  off  the  two  right  hand 
figures:  and, 

III.  To  reduce  mills  to  dollars :  cut  off  the  three  right 
hand  figures. 

EXAMPLES. 

1.  How  many  dollars  cents  and  mills  are  there  in  67897 
mills  ? 

2.  Set  down  104  dollars  69  cents  and  8  mills. 

3.  Set  down  4096  dollars  4  cents  and  2  mills. 

4.  Set  down  100  dollars  1  cent  and  1  mill. 

5.  Write  down  4  dollars  and  6  mills. 

6.  Write  down  109  dollars  and  1  mill. 

7.  Write  down  65  cents  and  2  mills. 

8.  Write  down  2  mills. 

9.  Reduce  1607  mills,  to  dollars  cents  and  mills. 
10.  Reduce  170464  mills,  to  dollars  cents  and  mills. 
IK  Reduce  8674416  mills,  to  dollars  cents  and  mills. 

12.  Reduce  94780900  mills,  to  dollars  cents  and  mills. 

13.  Reduce  74164210  mills,  to  dollars  cents  and  mills. 

8G.  How  do  you  change  mills  into  cents  ?    How  do  you  change  cento 
Into  dollars  ?    How  do  you  change  mills  to  dollars  ? 


UNITED   STATES  MONEY.  83 

87.  One  number  is  said  to  be  an  aliquot  part  of  another, 
when  it  is  contained  in  that  other  an  exact  number  of  times. 
Thus ;  50  cents,  25  cents,  &c.,  are  aliquot  parts  of  a  dollar : 
so  also  2  months,  3  months,.  4  months  and  6  months  are  ali- 
quot parts  of  a  year.  The  parts  of  a  dollar  are  sometimes 
expressed  fractionally,  as  in  the  following 

TABLE  OF  ALIQUOT  PARTS. 


$1  =100  cents. 

|  of  a  dollar  =  50  cents. 

|  of  a  dollar = 33 J  cents. 

J  of  a  dollar =  25  cents, 

of  a  dollar  =  20  cents. 


I    of  a  dollar^  121  cents. 

fa  of  a  dollar  =   10  cents. 

^  of  a  dollar =  6J  cents, 

z^j-  of  a  dollar  =     5  cents, 

of  a  cent    =     5  mills. 


ADDITION  OF  UNITED  STATES  MONEY. 

1.  Charles  gives  9|  cents  for  a  top,  and  3J  cents  for  6 
quills  :  how  much  do  they  all  cost  him  ? 

2.  John  gives  $1.37£  for  a  pair  of  shoes,  25  cents  for  a 
penknife,  and  12  J  cents  for  a  pencil :  how  much  does  he  pay 
for  all  ? 

OPERATION. 

ANALYSIS. — We  observe  that  half  a  cent  is  equal         $1.375 
to  5  mills.     We  then  place  the  mills,  cents  and  dol-  '25 

lars  in  separate  columns.    We  then  add  as  in  simple  I9f\ 

numbers.  •i-J° 

$1.750 

OPERATION. 

3.  James  gives  50  cents  for  a  dozen  oranges,         $0.50 
12|  cents  for  a  dozen  apples:  and  30  cents  for  .125 
a  pound  of  raisins  :  how  much  for  all  ?                           .30 

$0.925  ' 

88.  Hence,  for  the  addition  of  United  States  money,  we 
have  the  following 

RULE. — I.  Set  down  the  numbers  so  that  units  of  the 
same  value  shall  stand  in  the  same  column. 


87.  What  is  an  aliquot  part  ?    How  many  cents  in  a  dollar  ?    In  half 
a  dollar  ?    In  a  third  of  a  dollar  ?    In  a  fourth  of  a  dollar  ? 


84  APPLICATIONS   IN 

II.  Add  up  the  several  columns  as  in  simple  numbers, 
and  place  the  separating  point  in  the  sum  directly  under 
that  in  the  columns. 

PROOF. — The  same  as  in  simple  numbers. 

EXAMPLES. 

1.  Add  $61.214.  $10.049,  $6.041,  $0.271,  together. 

(1.)                                  (2.)  (3.) 

$  cts.  m.  $  cts.  m.  $  cts.  m. 

67.214  59.316  81.053 

10.049  87.425  67.412 

6.041  48.872  95.376 

0.271  56.708  87.064 

$83.575  $330.905 

APPLICATIONS. 

1.  A  grocer  purchased  a  box  of  candles  for  6  dollars 
89  cents  :  a  box  of  cheese  for  25  dollars  4  cents  and  3  mills  ; 
a  keg  of  raisins  for  1  dollar  12|  cents,  (or  12  cents  and  5 
mills ;)  and  a  cask  of  wine  for  40  dollars  37  cents  8  mills : 
what  did  the  whole  cost  him  ? 

2.  A  farmer  purchased  a  cow  for  which  he  paid  30  dollars 
and  4  mills ;  a  horse  for  which  he  paid  104  dollars  60  cents 
and  1  mill ;    a  wagon  for  which  he  paid  85  dollars   and 
9  mills  :  how  much  did  the  whole  cost  ? 

3.  Mr.  Jones  sold  farmer  Sykes  6  chests  of  tea  for  $75.641 ; 
9  yards  of  broadcloth  for  $27.41  ;  a  plow  for  $9.75 ;  and  a 
harness  for  $19.674  :  what  was  the  amount  of  the  bill  ? 

4.  A  grocer  sold  Mrs.  Williams  18  hams  for  $26.497  ;  a  bag 
of  coffee  for  $17.419  ;  a  chest  of  tea  for  $27.047  ;  and  a 
firkin  of  butter  for  $28.147  :  what  was  the  amount  of  her 
bill? 

5.  A  father  bought  a  suit  of  clothes  for  each  of  his  four 
boys  ;  the  suit  of  the  eldest  cost  $15.167  ;  of  the  second, 
$13.407  ;  of  the  third,  12.75  ;  and  of  the  youngest,  $11.047  : 
how  much  did  he  pay  in  all  ? 

88.  How  do  you  set  down  the  numbers  for  addition  ?  How  do  you 
add  up  the  columns  ?  How  do  you  place  the  separating  point  ?  How 
do  you  prove  addition  ? 


UNITED   STATES   MONEY.  85 

6.  A  father  has  six  children  ;  to  the  first  two  he  gives 
each  $375.416  ;  to  each  of  the  second  two,  $287.55  ;  to  each 
of  the  remaining  two,  $259.004  :  how  much  did  he  give  to 
them  all? 

7.  A  man  is  indebted  to  A,  $630.49  ;  to  B,  $25  ;  to  C, 
87  J  cents  ;  to  D,  4  mills  :  how  much  does  he  owe  ? 

8.  Bought  1  gallon  of  molasses  at  28  cents  per  gallon  ;  a 
half  pound  of  tea  for  78  cents ;  a  piece  of  flannel  for  12  dol- 
lars 6  cents  and  3  mills ;  a  plow  for  8  dollars   1  cent  and 

1  mill ;  and  a  pair  of  shoes  for  1  dollar  and  20  cents :  what 
did  the  whole  cost  ? 

9.  Bought  6  pounds  of  coffee  for  1  dollar  12J  cents  ;  a 
wash-tub  for  75  cents  6  mills  ;  a  tray  for  26  cents  9  mills  ;  a 
broom  for   27  cents  ;  a  box  of  soap  for  2  dollars  65  cents 
7  mills  ;  a  cheese  for  2  dollars  87^  cents  :  what  is  the  whole 
amount  ? 

10.  What  is  the  entire  cost  of  the  following  articles,  viz. : 

2  gallons   of  molasses,  57  cents ;  half  a  pound  of  tea,  37| 
cents  ;  2  yards  of  broadcloth,  $3.37|  cents  ;  8  yards  of  flan- 
nel, $9.875  ;  two  skeins  of  silk,  12|  cents,  and  4  sticks  of 
twist,  8i  cents  ? 

SUBTRACTION  OF  UNITED  STATES  MONEY. 

1.  John  gives  9  cents  for  a  pencil,  and  5' cents  for  a  top, 
how  much  more  does  he  give  for  the  pencil  than  for  the  top  ? 

2.  A  man   buys  a  cow  for  $26.37,  and  a  calf  for  $4.50  : 
how  much  more  does  he  pay  for  the  cow  than  for  the  calf  ? 

OPERATION. 

NOTE. — We  set  down  the  numbers  as  in  addition,       $26.37 
and  then  subtract  them  as  in  simple  numbers.  4  50 

$21.87 

89.  Hence,  for  subtraction  of  United  States  money,  we 
have  the  following 

RULE. — I.  Write  the  less  number  under  the  greater  so  thai 
units  of  the  same  value  shall  stand  in  the  same  column. 

89.  How  do  you  set  down  the  numbers  for  subtraction  ?  How  do 
you  subtract  them  ?  Where  do  you  place  the  separating  point  in  the 
remainder  ?  How  dc  you  prove  subtraction  ? 


86  SUBTRACTION  OF 

II.  Subtract  as  in  simple  numbers,  and  place  the  separating 
point  in  the  remainder  directly  under  that  in  the  columns. 

PROOF. — The  same  as  in  simple  numbers. 

EXAMPLES. 

(I-)  (2.) 

From              $204.679                 From  $8976.400 

Take                   98.714                 Take  610.098 
Remainder     $105.965                Remainder     $8366.302 

(3.)                     (4.)  (5.) 

$620.000             $327.001  $2349 

19.021                    2.090  29.33 

$600.979             $324.911  $2319.67 

6.  What  is  the  difference  between  $6  and  1  mill  ?  Between 
$9.75  and  8  mills  ?    Between  75  cents  and  6  mills?  Between 
$87.354  and  9  mills? 

7.  From  $107.003  take  $0.479. 

8.  From  $875.043  take  $704.987. 

9.  From  $904.273  take  $859.896. 

APPLICATIONS. 

1.  A  man's  income  is  $3000  a  year  ;  he  spends  $187.50  : 
how  much  does  he  lay  up  ? 

2.  A  man  purchased  a  yoke  of  oxen  for  $78,  and  a  cow  for 
$26.003 :  how  much  more  did  he  pay  for  the  oxen  than  for 
the  cow  ? 

3.  A  man  buys  a  horse  for  $97.50,  and  gives  a  hundred 
dollar  bill :  how  much  ought  he  to  receive  back  ? 

4.  How  much  must  be  added  to  $60.039  to  make  the  sum 
$1005.40? 

5.  A  man  sold  his  house  for  $3005,  this  sum  being  $98.039 
more  than  he  gave  for  it :  what  did  it  cost  him  ? 

6.  A  man  bought  a  pair  of  oxen  for  $100,  and  sold  th'em 
again  for  $7 5.37  J  :  did  he  make  or  lose  by  the  bargain,  and 
how  much  ? 

7.  A  man  starts   on   a  journey  with  $100 ;    he   spends 
$87.57  :  how  much  has  he  left? 

8.  How  much  must  you  add  to  $40.173  to  make  $100? 


UNITED   STATES   MONEY.  87 

9.  A  man  purchased  a  pair  of  horses  for  $450,  but  finding 
one  of  them  injured,  the  seller  agreed  to  deduct  $106.325  : 
what  had  he  to  pay  ? 

10.  A  farmer  had  a  horse  worth  $147.49,  and  traded  him 
for  a  colt  worth  but  $35.048  :  how  much  should  he  receive 
in  money  ? 

11.  My  house  is  worth  $8975.034;   my  barn  $695.879: 
what  is  the  difference  of  their  values  ? 

12.  What  is  the  difference  between  nine  hundred  and  sixty- 
nine  dollars  eighty  cents  and  1  mill,  and  thirty-six  dollars 
ninety-nine  cents  and  9  mills  ? 

MULTIPLICATION  OF  UNITED  STATES  MONEY. 

1.  John  gives  3  cents  apiece  for  6  oranges :  how  much  do 
they  cost  him  ? 

2.  John  buys  6  pairs  of  stockings,  for  which  he  pays  25 
cents  a  pair  :  how  much  do  they  cost  him  ? 

3.  A  farmer  sells  8  sheep  for  $1.25  each  :  how  much  does 
he  receive  for  them  ? 

OPERATION. 

ANALYSIS. — We  multiply  the  costs  of  one  sheep  by       $1.25 
the  number  of  sheep,  and  the  product  is  the  entire  '   o 

cost. 

$10.00 

90.  Hence,  for  the  multiplication  of  United  States  money 
by  an  abstract  number,  we  have  the  following 

RULE. — I.   Write  the  money  for  the  multiplicand,  and  the 
abstract  number  for  the  multiplier. 

II.  Multiply  as  in  simple  numbers,  and  the  product  will 
be  the  answer  in  the  lowest  denomination  of  the  multi- 
plicand. 

III.  Reduce  the  product  to  dollars,  cents  and  mills. 
PROOF. — Same  as  in  simple  numbers 

EXAMPLES. 
1.  Multiply  385  dollars  28  cents  and  2  mills,  by  8. 

OPERATION.  (2.) 

$385.282  $475.87 

8  9 

Product  $3082.256  Product  $4282.83 


88  MULTIPLICATION  OF 

3.  What  will  55  yards  of  cloth  come  to  at  37  cents  per 
yard? 

4.  What  will  300  bushels  of  wheat  come  to  at  $1.25  per 
bushel  ? 

5.  What  will  85  pounds  of  tea  come  to  at  1  dollar  37  £ 
cents  per  pound  ? 

6.  What  will  a  firkin  of  butter  containing  90  pounds  come 
to  at  25J  cents  per  pound  ? 

7.  What  is  the  cost  of  a  cask  of  wine  containing  29  gal- 
lons, at  2  dollars  and  75  cents  per  gallon  ? 

8.  A  bale  of  cloth  contains  95  pieces,  costing  40  dollars 
37  J  cents  each  :  what  is  the  cost  of  the  whole  bale  ? 

9.  What  is  the  cost  of  300  hats  at  3  dollars  and  25  cents 
apiece  ? 

10.  What  is  the  cost  of  9704  oranges  at  3J  cents  apiece  ? 

OPERATION. 

NOTE. — We  know  that  the  product  of  two  num- 
bers contains  the  same  number  of  units,  whichever 
be  used  as  the  multiplier  (Art.  48).  Hence,  we 
may  multiply  9704  by  3^  if  we  assign  the  proper 
unit  (1  cent)  to  the  product. 

$339.64 

11.  What  will  be  the  cost  of  356  sheep  at  3J  dollars  a 
head  ? 

12.  What  will  be  the  cost  of  47  barrels  of  apples  at  1  j 
dollars  per  barrel  ? 

13.  What  is  the  cost  of  a  box  of  oranges  containing  450, 
at  2  £  cents  apiece  ? 

14.  What  is  the  cost  of  307  yards  at  linen  of  68J  cents 
per  yard  ? 

15.  What  will  be  the  cost  of  65  bushels  of  oats  at  33*  cents 
a  bushel  ? 

ANALYSIS. — If  the  price  were  1  dollar  a  bushel,      OPERATION. 
the   cost  would  be  as  many  dollars  as  there  are        3)65.000 
bushels.     But  the  cost  is  38^  cents =£  of  a  dollar  :  .„.  flrpa 

hence,  the  cost  will  be  as  many  dollars  as  3  is  con- 
tained  times  in  65=21   dollars,  and  2  dollars  over,  which  is  re- 

90.  How  do  you  multiply  United  States  money  ?  What  will  be  the 
denomination  of  the  product  ?  How  will  you  then  reduce  it  to  dollars 
and  cents  ?  How  do  you  prove  multiplication  ? 


UNITED   STATES  MONEY.  89 

duced  to  cents  by  annexing  two  ciphers,  and  to  mills  by  annexing 
three ;  then,  dividing  the  cents  and  mills  by  3,  we  have  the  entire 
cost:  hence, 

91.  To  find  the  cost,  when  the  price  is  an  aliquot  part  of 
a  dollar. 

Take  such  a  part  of  the  number  which  denotes  the  commo- 
dity, as  the  price  is  of  I  dollar. 

EXAMPLES. 

1.  What  would  be  the  cost  of  345  pounds  of  tea  at  50 
cents  a  pound  ? 

2.  What  would  675  bushels  of  apples  cost  at  25  cents  a 
bushel  ? 

3.  If  1  pound  of  butter  cost  12|  cents,  what  will  4  firkins 
cost,  each  weighing  56  pounds  ? 

4.  At  20  cents  a  yard,  what  will  42  yards  of  cloth  cost  ? 

5.  At  33  J  cents  a  gallon,  what  will  136  gallons  of  mo- 
lasses cost  ? 

OPERATION. 

6.  What  will  1276  yds.     4)$1276  cost  at  1  dollar  a  yard, 
of  cloth  cost  at  $1.25  a  319  cost  at  25  cts.  a  yard, 
yard  ?                                         $1595  Cost  at  $1.25  a  yard. 

7.  What  would  be  the  cost  of  318  hats  at  $1.12J  apiece  ? 

8.  What  will  2479  bushels  of  wheat  come  to  at  $1.50 
a  bushel  ? 

9.  At  $1.33J  a  foot,  what  will  it  cost  to  dig  a  well  78  feet 
deep  ? 

10.  What  will  be   the  cost  of  936  feet  of  lumber  at  3 
dollars  a  hundred  ? 

ANALYSIS. — At  3  dollars  a  foot  the  cost  would  be  OPERATION. 
936x3=2808  dollars ;  but  as  3  dollars  is  the  price  935 

of  100  feet,  it  follows  that  2808  dollars  is  100  times 

the  cost  of  the   lumber:    therefore,   if  we  divide          

2808  dollars  by  100  (which  we  do  by  cutting  off  two         $28.08 
of  the  right  hand  figures  (Art.  73),  we  shall  obtain  the  cost. 

NOTE. — Had  the  price  been  so  much  per  thousand,  we  should 
have  divided  by  1000,  or  cut  off  three  of  the  right  hand  figures : 
hence, 

91.  How  do  you  find  the  cost  of  several  things  when  the  price  is  an 
aliquot  part  of  a  dollar  ? 


90  MULTIPLICATION   OF 

92.  To  find  the  cost  of  articles  sold  by  the  100  or  1000  ; 

Multiply  the  quantity  by  the  price  ;  and  if  the  price  be 
by  the  100,  cut  off  two  figures  on  the  right  hand  of  the 
product ;  if  by  the  1000,  cut  off  three,  and  the  remaining 
figures  will  be  the  answer  in  the  same  denomination  as  the 
price,  which  if  cents  or  mills,  may  be  reduced  to  dollars. 

EXAMPLES. 

1.  What  will  4280  bricks  cost  at  $5  per  1000  ? 

2.  What  will  2673  feet  of  timber  cost  at  $2.25  per  100  ? 

3.  What  will  be  the  cost  of  576  feet  of  boards  at  $10.62 
per  1000  ? 

4.  What  is  the  value  of  1200  feet  of  lathing  at  7  dollars 
per  1000  ? 

5.  David  Trusty,  Bought  of  Peter  Bigtree. 
2462  feet  of  boards    at  $7.       per  1000. 


4520 

«                    u 

'     9.50 

600 

"     scantling 

1  11.37 

960 

"     timber 

1  15. 

1464 

"     lathing 

.75   per  100. 

1012 

"     plank 

'     1.25 

Received  Payment, 

Peter  Bigtree, 

6.  What  is  the  cost  of  1684  pounds  of  hay  at  $10.50  per 
ton? 

ANALYSIS. — Since   there    are  OPERATION. 

2000*.    in  a    ton,  the    cost  of  2)10.50 

?o°r00"  ^$5™^  ~5£5  price  of  1000ftS. 

cents.      Multiply    this    by  the  1684 

number  of  pounds  (1684),  and          $g  841QO  Ans. 

cut   off  three   places   from  the 

right,,  in  addition  to  the  two  places  before  cut  off  for  cents :  hence, 

93.  To  find  the  cost  of  articles  sold  by  the  ton  : 
Multiply  one-half  the  price  of  a  ton  by  the  number  of 
pounds' and  cut  off  three  figures  from  the  right  hand  of 
the  product.     The  remaining  figures  will  be  the  answer  i 

the  same  denomination  as  the  price  of  a  ton. 

92.  How  do  you  find  the  cost  of  articles  sold  by  the  100  or  1000  ? 


UNITED   STATES  MONEY.  91 

EXAMPLES. 

1.  What  will  3426  pounds  of  plaster  cost  at  $3.48  per  ton? 

2.  What  will  be  the  cost  of  the  transportation  of  6742 
pounds  of  iron  from  Buffalo 'to  New  York,  at  $7  per  ton  ? 

3.  What  will  be  the  cost  of  840  pounds  of  hay  at  $9.50 
per  ton?  at  $12?  at  $15.84  ?  at  $10.36  ?  at  $18.75? 

DIVISION  OP  UNITED  STATES  MONEY. 

94.  To  divide  a  number  expressed  in  dollars,  cents  or  mills, 
into  any  number  of  equal  parts. 

RULE. — I.  Reduce  the  dividend  to  cents  or  mills,  if  necessary. 

II.  Divide  as  in  simple  numbers,  and  the  quotient  will  be  the 
answer  in  the  lowest  denomination  of  the  dividend :  this  may 
be  reduced  to  dollars,  cents,  and  mills. 

PROOF. — Same  as  in  division  of  simple  numbers. 

NOTE. — The  sign  +  is  annexed  in  the  examples,  to  show  that 
there  is  a  remainder,  and  that  the  division  may  be  continued. 

EXAMPLES. 

1.  Divide  $4.624  by  4  :  also,  $87.256  by  5. 

OPERATION.  OPERATION. 

4)$4.624  5j$87.256 


$1.156  $17.454 

2.  Divide  $37  by  8. 

ANALYSIS. — In  this  example  we  first  reduce  the  OPERATION. 

$37  to  mills  by  annexing  three  ciphers.     The  quo-  8)$37,000 

tient  will  then  be  mills,  and  can  be  reduced  to  dol-  — <fe   //fio^ 

lars  and  cents,  as  before.  v  4,bJo 

3.  Divide  $56.16  by  16. 

4.  Divide  $495.704  by  129. 

5.  Divide  $12  into  200  equal  parts. 

6.  Divide  $400  into  600  equal  parts. 

7.  Divide  $857  into  51  equal  parts. 

8.  Divide  $6578.95  into  157  equal  parts. 

93.  How  do  you  find  the  cost  of  articles  sold  by  the  ton  ? 

94.  What  is  the  rule  for  division  of  United  States  money  ?    How  do 
you  prove  division  ?    How  do  you  indicate  that  the  division  may  be 
continued  ? 


92  DIVISION   OF 

95.  The  quantity,  and  the  cost  of  a  quantity  given,  to  find 
the  price  of  one  thing  (Art.  80). 

Divide  the  cost  by  the  quantity. 

9.  Bought  9  pounds  of  tea  for  $5.85  ;  what  was  the  price 
per  pound  ? 

10.  Paid  $29.68  for  14  barrels  of  apples:  what  was  the 
price  per  barrel  ? 

11.  If  27  bushels  of  potatoes  cost  $10.125,  what  is  the 
price  of  a  bushel  ? 

12.  If  a  man  receive  $29.25  for  a  month's  work,  how 
much  is  that  a  day,  allowing  26  working  days  to  the  month  ? 

13.  A  produce  dealer  bought  3  barrels  of  eggs,  each  con- 
taining 150  dozens,  for  which  he  paid  $63 :  how  much  did 
he  pay  a  dozen  ? 

14.  A  man  bought  a  piece  of  cloth  containing  72  yards, 
for  which  he  paid  $252  :  what  did  he  pay  per  yard  ? 

15.  If  $600  be  equally  divided  among  26  persons,  what 
will  be  each  one's  share  ? 

16.  Divide  $18000  into  40  equal  parts:  what  is  the  value  of 
each  part  ? 

17.  Divide  $3769.25   into  50  equal  parts:    what  is  one 
part? 

18.  A  farmer  purchased  a  farm  containing  725  acres,  for 
which  he  paid  $18306.25  :  what  did  it  cost  him  per  acre  ? 

19.  A  merchant  buys  15  bales  of  goods  at  auction,  for 
which  he  pays  $1000  :  what  do  they  cost  him  per  bale  ? 

20.  A  drover  pays  $1250  for  500  sheep ;  what  shall  he 
sell  them  for  apiece,  that  he  may  neither  make  nor  lose  by 
the  bargain  ? 

21.  The  dairy  of  a  farmer  produces  $600,  and  he  has  25 
cows  :  how  much  does  he  make  by  each  cow  ? 

22.  A  farmer  receives  $840  for  the  wool  of  1400  sheep  : 
how  much  does  each  sheep  produce  him  ? 

23.  A  merchant  buys  a  piece  of  goods  containing   105 
yards,  for  which  he  pays  $262.50  ;  he  wishes  to  sell  it  so  as 
to  make  $52.50  :  how  much  must  he  ask  per  yard? 

90.  When  the  price  of   one  and  the  cost  of  a  quantity  are 
given,  to  find  the  quantity  (Art.  80). 

Nora—The  divisor  and  dividend  must  both  be  reduced  to  the 
lowest  unit  named  in  either  before  dividing. 


UNITED   STATES    MONEY.  93 

Divide  the  cost  by  the  price. 

24.  If  I  pay  $4.50  a  ton  for  coal,  how  much  can  I  buy 
for  $67.50  ? 

25.  At  $7  a  barrel,  how  much  flour  can  be  bought  for 
$178.50? 

26  How  many  pounds  of  tea  can  be  bought  for  $6.75,  at 
75  cents  a  pound  ? 

27.  What  number  of  barrels  of  apples  can  be  bought  for 
$47.50,  at  $2.37 J  a  barrel? 

28.  At  44  cents  a  bushel,  how  many  bushels  of  oats  can 
be  bought  for  $14.30  ? 

29.  At  34  cents  a  bushel,  how  many  barrels  of  apples  can 
I  buy  for  $13.60,  allowing  2J  bushels  to  the  barrel? 

30.  If  1  acre  of  land  costs  $28.75,  how  much  can  be 
bought  for  $3220  ? 

31.  Paid  $40.50  for  a  pile  of  wood,  at  the  rate  of  $3.37J 
a  cord,  how  much  was  there  in  the  pile  ? 

32.  How  many  sheep  can  be  bought  for  $132,  at  $1.37|  a 
head  ? 

33.  At  $4.25  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $68  ? 

34.  At  $1.12J  a  day,  how  long  would  it  take  a  person  to 
earn  $157.50. 

APPLICATIONS  IN  THE  FOUR  PRECEDING  RULES. 

NOTE. — See  and  repeat  Rule — page  53 :  also  the  three  rules — 
page  74. 

1.  If  1  yard  of  cloth  costs  3  J  dollars,  what  will  8  yards  cost  ? 

2.  If  1  ton  of  hay  costs  $14  J,  what  will  9  tons  cost  ? 

3.  If  1  calf  costs  $4  J,  what  will  12  calves  cost  ? 

4.  Mr.  Jones  bought  250  bushels  of  oats,  for  which  he  paid 
$156.25  :  how  much  did  they  cost  him  a  bushel  ? 

5.  If  12  tons  of  hay  cost  150  dollars,  what  does  1  ton 
cost  ?  8  tons  ?  50  tons  ? 

6.  If  9  dozen  of  spelling  books  cost  $7.875,  what  will  1 
dozen  cost  ?  6  dozen  ?  8  dozen  ? 

7.  If  75  bushels  of  wheat  cost  $131.25,  how  much  will  1 
bushel  cost  ?  8  bushels  ?  120  bushels  ? 

8.  If  320  pounds  of  coffee  cost  $44.80  cents,  how  much 
will  1  pound  cost  ?    What  will  575  pounds  cost  ? 


94:  APPLICATIONS  IN 

9.  Mr.  James  B.  Smith  bought  9  barrels  of  sugar,  each 
weighing  216  pounds,  for  which  he  paid  $116.64  :  how  much 
did  he  pay  a  pound  ? 

10.  If  40  tons  of  hay  cost  $580,  how  much  is  that  per 
ton  ?     What  would  70  tons  cost  at  the  same  rate  ? 

11.  If  Mr.  Wilson  has  $120  to  buy  his  winter  wood,  and 
wood  is  $4  a  cord,  how  many  cords  can  he  buy  ? 

12.  At  6  dollars  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $24  ?     How  many  for  $36  ? 

13.  A  farmer  sold  a  yoke  of  oxen  for  $80.75  ;  6  cows  for 
$29  each ;  30  sheep  at  $2.50  a  head ;  and  3  colts,  one  for 
$25,  the  other  two  for  $30  apiece ;  what  did  he  receive  for 
the  whole  lot  ? 

14.  A  merchant  buys  6  bales  of  goods,  each  containing  20 
pieces  of  broadcloth,  and  each  piece  of  broadcloth  contained 
29  yards  ;  the  whole  cost  him  $15660  ;  how  many  yards  of 
cloth  did  he  purchase,  and  how  much  did  it  cost  him  per 
yard? 

15.  A  person  sells  3  cows  at  $25  each  ;  and  a  yoke  of 
oxen  for  $65  ;  he  agrees  to  take  in  payment  60  sheep  :  how 
much  do  his  sheep  cost  him  per  head  ? 

16.  A  man  dies  leaving  an  estate  of  $33000  to  be  equally 
divided  among  his  4  children,  after  his  wife  shall  have  taken 
her  third.     What  was  the  wife's  portion,  and  what  the  part 
of  each  child  ? 

17.  A  person  settling  with  his  butcher,  finds  that  he  is 
charged  with  126  pounds  of  beef  at  9  cents  per  pound  ;  85 
pounds  of  veal  at  6  cents  per  pound  ;  6  pairs  of  fowls  at  37 
cents  a  pair  ;  and  three  hams  at  $1,50  each  :  how  much 
does  he  owe  him  ? 

18.  A  farmer  agrees  to  furnish  a  merchant  40  bushels  of 
rye  at  62  cents  per  bushel,  and  to  take  his  pay  in  coffee  at 
16  cents  per  pound  :  how  much  coffee  will  he  receive  ? 

19.  A  farmer  has  6  ten-acre  lots,  in  each  of  which  he  pas- 
tures 6  cows  ;  each  cow  produces  112  pounds  of  butter,  for 

1  which  he  receives  18 \  cents  per  pound  ;  the  expenses  of 
each  cow  are  5  dollars  and  a  half :  how  much  does  he  make 
by  his  dairy  ? 

20.  Bought  a  farm  of  W.  N.  Smith  for  2345  dollars,  a 
span  of  horses  for  375  dollars,  6  cows  at  36  dollars  each  ?  I 
paid  him  520  dollars  in  cash,  and  a  village  lot  worth  1500 
dollars :  how  many  dollars  remain  unpaid  ? 


UNITED  STATES  MONEY.  95 

BILLS    OF    PARCELS. 

(21.)  New  York,  May  1st,  1854. 

Mr.  James  Spendthrift, 

Bought  of  Benj.  SavedLl. 

16  pounds  of  tea  at  85  cents  per  pound  -  -  - 
27  pounds  of  coffee  at  15J  cents  per  pound  -  - 
15  yards  of  linen  at  66  cents  per  yard  -  -  -  - 


Received  payment,  Benj.  Saveall. 

(22.)  Albany,  June  2d,  1854; 

Mr.  Jacob  Johns, 

Bought  of  Gideon  Gould. 

36  pounds  of  sugar  at  9  J  cents  per  pound     -    - 
3  hogsheads  of  molasses,  63  galls,  each,  at  27 

cents  a  gallon 

5  casks  of  rice,  285  pounds  each,  at  5  cents  per 

pound      

2  chests  of  tea,  86  pounds  each,  at  96  cents  per  ) 

pound f 

Total  cost,     $ 
Received  payment,        For  Gideon  Gould, 

Charles  Clark. 


<J23.)  Hartford,  November  21st,  1854. 

Gideon  Jones, 

Bought  of  Jacob  Thrifty. 

69  chests  of  tea  at  $55.65  per  chest  -    -    -    - 
126  bags  of  coffee,  100  pounds  each,  at  12J ) 

cents  per  pound } 

167  boxes  of  raisins  at  $2.75  per  box     -    -    - 

800  bags  of  almonds  at  $18.50  per  bag  -    -    - 

9004  barrels  of  shad  at  $7.50  per  barrel  -    -    - 

60  barrels  of  oil,  32  gallons  each,  at  $1.08 ) 

per  gallon ) 

Amount,     $ 
Received  the  above  in  full.  Jacob  Thrifty. 


90  DENOMINATE  NUMBERS. 


DENOMINATE    NUMBERS. 

97.  A  SIMPLE  NUMBER  is  a  unit  or  a  collection  of  units. 
The  unit  may  be  either  abstract  or  denominate. 

98.  A  DENOMINATE   NUMBER  is  a  denominate   unit  or   a 
collection  of  units  :  thus,  3  yards  is  a  denominate  number, 
in  which  the  unit  is  1  yard. 

99.  Numbers  which  have  the  same  unit,  are  of  the  same 
denomination:  and  numbers  having  different  units,  are  of 
different  denominations.     If  two  or  more  denominate  num- 
bers, having  different  units,  are  connected  together,  forming  a 
single  number,  such  is  called  a  compound  denominate  number. 

100.  There  are  eight  different  units  in  Arithmetic  :  1st. 
The  abstract  unit :  2d.  The  unit  of  currency  :  3d.  The  unit 
of  length  :  4th.  The  unit  of  surface  :  5th.  The  cubic  unit  or 
unit  of  volume  :  6th.  The  unit  of  weight :  7th.  The  unit  of 
time :  8th.  The  unit  of  circular  measure. 

ENGLISH  MONEY. 

101.  The  units  or  denominations  of  English  money  are 
guineas,  pounds,  shillings,  pence,  and  farthings. 

TABLE. 

4  farthings  marked  far  make  1  penny,     marked       d. 
12  pence  -  1  shilling,  s. 

20  shillings       -  1  pound,  or  sovereign,  £, 

21  shillings       -  -     .   1  guinea. 

far.  d.  s.  £ 

4  =1 

48  =12  =  1 

960  =240  =20  =1 

NOTES. — 1.  The  primary  unit  in  English  money  is  1  farthing. 
The  number  of  units  in  the  scale,  in  passing  from  farthings  to 


97.  What  is  a  simple  number  ? 

98.  What  is  a  denominate  number  ? 

99.  When  are  numbers  of  the  same  denomination  ?    When  of  differ- 
ent denominations  ?    If  several  numbers  having  different  units  are  con- 
nected together,  what  is  the  number  called  ? 

100.  How  many  units  are  there  in  Arithmetic  ?    Name  them, 


DENOMINATE   NUMBERS.  97 

pence,  is  4  ;    in  passing  from  pence  to  shillings,  12  ;    in  passing 
from  shillings  to  pounds,  20. 

2.  Farthings  are  generally  expressed  in  fractions  of  a  penny. 
Thus,  1  far.=tf.;  2  far.=\d.  ;  3  far.=$d. 

3.  By  reading   the  second  table  from  right  to  left,  we  can  see 
the  value  of  any  unit  expressed  in  each  of  the  lower  denomina- 
tions.     Thus,    ld.  =  4far.;     1*.=  12d.=4Stfar.  ;    £l=20«.=  240d. 


REDUCTION  OF  DENOMINATE  NUMBERS. 

102.  Reduction  is  changing  the  unit  of  a  number,  without 
altering  its  value. 

1.  How  many  pence  are  there  in  2s.  &d.  ? 

ANALYSIS.  —  Since  there  are  12  pence  in  1  shilling,  there  are 
twice  12,  or  24  pence  in  2  shillings  :  add  the  6  pence  :  therefore, 
in  2s.  6d.  there  are  30  pence. 

2.  How  many  pence  in  4  shillings?     In  4s.  Sd.  ?     In  5s. 
Sd.  ?     In  3s.  Sd.  ?     In  6s.  Id.  ? 

3.  How  many  shillings  in  ,£2  ?     In  £3  8s.,  how  many  ? 

4.  How  many   pence   in   £1  ?      How  many  shillings   in 
£2  8s.  ?     How  many  in  ^3  7s.  ? 

5.  How  many  shillings  are  there  in  48  pence  ? 

ANALYSIS.  —  Since  there  are  12  pence  in  1  shilling,  there  are  as 
many  shillings  in  48  pence,  as  12  is  contained  times  in  48,  which 
is  4:  therefore,  there  are  4  shillings  in  48  pence. 

6.  How  many  pounds  in  40  shillings  ?     In  60  ?     In  80  ? 

103.  From  the  above  analyses  we  see,  that  reduction  of 
denominate  numbers  is  divided  into  two  parts  : 

1st.   To  change  the  unit  of  a  number  from  a  higher  deno- 
mination to  a"  lower. 

2d.   To  change  the  unit  of  a  number  from  a  lower  denomi- 
nation to  a  higher. 

101.  What  are  the  denominations  of  English  money  ? 

Notes.  1  —  What  is  the  primary  unit  in  English  money  ?  Name  the 
units  of  the  scale. 

2.  —  How  are  farthings  generally  expressed  ? 
3.—  How  is  the  second  table  read  ?    What  does  it  show  ? 

102.  What  is  Reduction  ? 

103.  Into  how  many  parts  is  reduction  divided  ?    What  are  tliey  ? 

7 


98  REDUCTION  OF 

PRINCIPLES   AND   EXAMPLES. 

104.  To  reduce  from  a  higher  to  a  lower  unit. 
1.  Reduce  JE21  6s.  Sd.  to  the  denomination  of  farthings 

OPERATION. 

ANALYSIS.—  Since  there  are  20  shillings  in    £27  6s   &d  2far 
£1,  in  £27  there  are  27  times  20  shillings,    '      on    ' 
or  540  shillings,  and  6  shillings  added,  make 
546*.     Since   12   pence  make   1   shilling,  we 
next  multiply  by  12,  and  then  add  Sd.  to  the 
product,    giving  6560    pence.      Since  4  far-     pf.Rn  , 
things  make  1   penny,  we  next  multiply  by 
4,  and   add  2  farthings  to  the  product,  giv- 
ing 26242  farthings  for.  the  answer.  26242 


NOTE.  —  The  units  of  the  scale,  in  passing  from  pounds  to  shil- 
lings, are  20  ;  in  passing  from  shillings  to  pence  they  are  12  ; 
and  in  passing  from  pence  to  farthings,  4. 

Hence,  to  reduce  from  a  higher  to  a  lower  unit,  -we  have 
the  following 

RULE.  —  Multiply  the  highest  denomination  by  the  units  of 
the  scale  which  connect  it  with  the  next  lower,  and  add  to  the 
product  the  units  of  that  denomination  ;  proceed  in  the  same 
manner  through  all  thd  denominations,  till  the  unit  is  brought 
to  the  required  denomination. 

105.   To  reduce  from  a  lower  unit  to  a  higher. 

1.  Reduce  3138  farthings  to  pounds. 

OPERATION. 

ANALYSIS.  —  Since    4     farthings         4)3138 
make   a  penny,  we  first  divide  by  4.          1  0N^Q  _   « 

Since   12  pence  make  a  shilling,  we  _  '  2Jar-  rCTn- 

next    divide   by   12.     Since  20  shil-  210)615  -  -    4d.  rem. 

lings  make  a  pound,  we  next  divide  c    "  r      ' 

by  20,   and    find  that   §l38/ar.=£3       -  —  " 
5s.  4d.  2  far.  Ans.  £3  5s.  4d.  2  far. 

Hence,  to  reduce  from  a  lower  to  a  higher  denomination, 
•we  have  the  following 

RULE.  —  I.  Divide  the  given  number  by  the  units  of  the  scale 

104.  How  do  you  reduce  from  a  higher  to  a  lower  unit? 

105.  How  do   you  reduce  from  a  lower  to  a  higher  unit?     What 
will  be"  the  unit  of  any  remainder  ?    How  do  you  prove  reduction  ? 


DENOMINATE   NUMBERS.  99 

which  connect  it  with  the  next  higher  denomination,  and  set 
down  the  remainder,  if  there  be  one. 

II.  Divide  the  quotient  thus  obtained  by  the  units  of  the 
scale  which  connect  it  with  the  next  higher  denomination,  and 
set  down  the  remainder. 

III.  Proceed  in  the  same  way  to  the  required  denomination, 
and  the  last  quotient,  with  the  several  remainders  annexed, 
ivill  be  the  answer. 

NOTE. — Every  remainder  will  be  of  the  same  denomination  as 
its  dividend. 

PROOF. — After  a  number  has  been  reduced  from  a  higher 
denomination  to  a  lower,  by  the  first  rule,  let  it  be  reduced 
back  by  the  second  ;  and  after  a  number  has  been  reduced 
from  a  lower  denomination  to  a  higher,  by  the  second  rule, 
let  it  be  reduced  back  by  the  first  rule.  If  the  work  is  right, 
the  results  will  agree. 

EXAMPLES. 

1.  Reduce  £15  7s.  &d.  to  pence. 

OPERATION.  PROOF. 

£15  7s.  Gd.  12)3690 

20  2|0)30|7  ...  6^.  rem. 

307  15      .  .  .  7s.  rem. 
12 

3690  Ans.  £15  7s.  bd. 

2.  In  £31  8s.  9<1  3  far.,  how  many  farthings?  Also  proof. 

3.  In  £87  14s.  8^d.,  how  many  farthings  ?     Also  proof. 

4.  In  £407  19s.  11  %d.,  how  many  farthings?     Also  proof. 

5.  In  80  guineas,  how  many  pounds  ? 

6.  In  1549  far.,  how  many  pounds,  shillings  and  pence? 

7.  In  6169  pence,  how  many  pounds  ? 

LINEAR  MEASURE. 

100.  This  measure  is  used  to  measure  distances,  lengths, 
breadths,  heights  and  depths,  &c. 

106.  For  what  is  Linear  Measure  used  ?  What  are  its  denominations  ? 
Repeat  the  table.  What  is  a  fathom?  What  is  a  hand?  What  are 
the  units  of  the  scale  ? 


100 


REDUCTION   OF 


TABLE. 


12  inches,  in.        make 
3  feet 

5J  yards  or  16  J  feet  - 
40  rods      - 
8  furlongs  or  320  rods 
3  miles 

69J  statute  miles  (nearly)  or 
60  geographical  miles, 
360  degrees, 

ft. 


1  foot, 
1  yard, 
1  rod, 
1  furlong,     - 
1  mile, 
1  league, 
1  degree  of) 

marked 

Af>.n.  i 

a 

rd. 
fur. 
mi. 
L. 

™  ° 

n. 
12 
36 
198 
7920 


=3 

=  16 
=  66 


yd. 
=1 


=  220 


the  equator 
a  circum'nce  of  the  earth. 
rd. 


63360       =  5280       =  1760 


=  1 

=  40 
=  320 


fur. 


_i_t     •• 

=  8 


mi. 


NOTES. — 1.  A  fathom  is  a  length  of  six  feet,  and  is  generally 
Bed  to  measure  the  depth  of  water. 

2.  A  hand  is  4  inches,  used  to  measure  the  height  of  horses. 

3.  The  units  of  the  scale,  in  passing  from  inches  to  feet,  are  12  ; 
in  passing  from  feet  to  yards,  3 ;    from  yards  to  rods,  5£ ;   from 
rods  to  furlongs,  40  ;  and  from  furlongs  to  miles,  8. 

1.  How  many  inches  in  5  feet  ?     In  10  feet  ?     In  16  feet  ? 

2.  How  many  yards  in  36  feet  ?     In  54  feet  ?     In  96  ? 

3.  How  many  feet  in  144  inches  ?     In  96  inches  ?     In  48  ? 

4.  How  many  furlongs  in  3  miles  ?     In  6  miles  ?     In  8  ? 


EXAMPLES. 


1.  How  many  inches  in 
&rd.  4yd.  2ft.  9in. 

OPERATION. 

6rdL  4yd.  2ft.  9in. 

_M 

3 

34 


37  yards. 
3 

113  feet. 
12 
1365  inches. 


2.  In  1365   inches,  how 

many  rods  ? 


OPERATION. 

12)1365 

3)113  feet    9m. 
5|)37  yards  2ft 
11)74 
6rd. 


Ans.  Qrd.  4yd.  2ft.  9m. 


DENOMINATE  NUMBERS. 


101 


NOTE. — When  we  reduce  rods  to  yards,  we  multiply  by  the 
scale  5i ;  that  is,  we  take  6  rods  5  and  one-half  times.  When  we 
reduce  yards  to  rods,  we  divide  by  5i,  which  is  done  by  reducing 
the  dividend  and  divisor  to  halves :  the  remainder  is  8  half-yards, 
equal  to  4  yards. 

3.  In  59wi.  *lfur.  38rY?.,  how  many  feet  ? 

4.  In  115188  rods,  how  many  miles? 

5.  In  719??u'.  I6rd.  6yd.,  how  many  feet? 
(6.  In  118°,  how  many  miles? 

7.  In  54°  45mi.  7/ur.  20rd.  ±yd.  2ft.  Win.,  how  many 
Inches  ? 

8.  In  481401716  inches,  how  many  degrees,  &c.  ? 

CLOTH  MEASURE. 

107.  Cloth  measure  is  used  for  measuring  all  kinds  of 
cloth,  ribbont;,  and  other  things  sold  by  the  yard. 

TABLE. 

nail,  marked  na. 
quarter  of  a  yard,  qr. 
Ell  Flemish,  E.  Fl. 

yard,  -     yd. 

Ell  English,      ,     E.  E. 


2J  inches,  in. 
4    nails 

make     1 
1 

3    quarters  - 
4    quarters  - 
5    quarters  - 

1 
1 

1 

in.             na. 
2J       —     1 

qr. 

9         =4 

=  1 

27          =  12 

=  3 

36         =  16 

=  4 

45         =  20 

=  5 

E.Fl 


=  1 


yd. 


-  l 


E.  E. 


=  1 


NOTE. — The  units  in  this  measure  are,  inches,  nails,  quarters,  Klls 
Flemish,  yards,  and  Ells  English. 

1.  In  9  inches,  how  many  nails  ?     How  many  nails  in  1 
yard  ?     In  2  yards  ?     In  6  ?     In  8  ? 

2.  In  4  yards,  how  many  quarters  ?     How  many  quarters 
in  8  yards  ?     In  7  how  many  ? 

3.  How  many  quarters  in  12  nails?     In  16  nails?     In  20 
nails?     In  36?     In  40  ? 


107.  For  what  is  cloth  measure  used  ?    What  are  its  denominations  ? 
Repeat  the  table.     What  are  the  units  of  this  measure  ? 


102 


REDUCTION  OF 


1. 


How  many  nails   are 
there  in  35yd.  3^r.  3na.  ? 

OPEKATION. 

35t/d.  3(? 
4 


EXAMPLES. 


143  quarters. 
4 


575  nails. 


2.  In   575   nails,   how 
many  yards  ? 

OPERATION. 

4)575 


4)143  3na. 
35  3  jr. 


Ans. 


.  3gr. 


3.  In  49  E.  E.,  how  many  nails  ? 

4.  In  51  i?.  FL,  2qr.  8na.,  how  many  nails  ? 

5.  In  3278  nails,  how  many  yards  ? 

6.  In  340  nails,  how  many  Ells  Flemish  ? 

7.  In  4311  inches,  how  many  E.  E.  ? 

SQUARE  MEASURE. 

108.  Square  measure  is  used  in  measuring  land,  or  anything 
in  which  length  and  breadth  are  both  considered. 

1  Foot. 

A  square  is  a  figure  bounded  by  four  equal 
lines  at  right  angles  to  each  other.  Each 
line  is  called  a  side  of  the  square.  If  each 
side  be  one  foot,  the  figure  is  called  a 
square  foot. 

If  the  sides  of  the  square  be  each  one 
yard,  the  square  is  called  a  square  yard. 
In  the  large  square  there  are  nine  small 
squares,  the  sides  of  which  are  each  one 
foot.  Therefore,  the  square  yard  contains 
9  square  feet. 

The  number  of  small  squares  that  is  contained  in  any  large 
square  is  always  equal  to  the  product  of  two  of  the  sides  of  the 
large  square.  As  in  the  figure,  3  x3~9  square  feet.  The  number 
of  square  inches  contained  in  a  square  foot  is  equal  to  12  x  12=144. 

108.  For  what  is  Square  Measure  used?  What  is  a  square?  If 
each  side  be  one  foot,  what  is  it  called  ?  If  each  side  be  a  yard,  whnt 
is  it  called  ?  How  many  square  feet  docs  the  square  yard  contain  ? 
How  is  the  number  of  small  squares  contained  in  a  large  square  found  ? 
Repeat  the  table.  What  are  the  units  of  the  scale  ? 


DENOMINATE    NUMBERS. 


103 


TABLE. 

144  square  inches,  sq.  in.,  make  1  square  foot, 

9  square  feet 

30  J  square  yards      - 

40  square  rods  or  perches   - 

4  roods     - 

640  acres     - 


Sq.ft. 

I  square  yard,  Sq.  yd. 

1  square  rod  or  perch,     P. 
1  rood,     -  E. 

1  acre,     -  A. 

1  square  mile,  M. 


Sq.  in. 

144 

1296 

39204 

1568160 

6272640 


_Sq.ft. 

=  9 
=  272J 
=  10890 
=  43560 


Sq.  yd. 

1 

301 

1210 

4840 


P. 


1 

40 
160 


E. 


=  1 

=  4  =1. 


NOTE.— The  uDits  of  the  scale  are  144,  9,  30L  40,  4  and  640. 

1.  How  many  square  inches  in  2  square  feet?     How  many 
square  feet  in  3  square  yards  ?     How  many  in  6  ?     In  8  ? 

2.  How  many  perches  in  1  rood  ?  In  3  roods  ?    How  many 
roods  in  4  acres  ?     In  8  ?     In  12  ? 

3.  How  many  perches  in  an  acre  ?     How  many  in  2  acres  ? 
How  many  square  yards  in  81  square  feet? 

SURVEYORS'  MEASURE. 

109.  The  Surveyor's  or  Gunter's  chain  is  generally  used  in 
surveying  land.  It  is  4  poles  or  66  feet  in  length,  and  is 
divided  into  100  links. 


inches         make 

4  .rods  or  66/X 
80  chains - 

1  square  chain 
10  square  chains 


TABLE. 

1  link,  marked  -  I. 

1  chain,     -  c. 

I  mile,       -  mi. 

16  square  rods  or  perches,  P. 

1  acre,        -  A. 


NOTE. — 1.  Land  is  generally  estimated  in  square  miles,  acres> 
roods,  and  square  rods  or  perches. 
2.  The  units  of  the  scale  are  7f9o20-,  4,  80. 


109.  What  chain  is  used  in  land  surveying  ?  What  is  its  length  ? 
How  is  it  divided?  Repeat  the  table.  In  what  is  land  generally  esti- 
mated ?  What  are  the  units  of  the  scale  ? 


104  REDUCTION   OF 

1.  How  many  rods  in  1  chain  ?     How  many  in  4  ?     In  5  ? 

2.  How  many  chains  in  1  mile  ?     In  2  miles  ?     In  3  ? 

3.  How  many  perches  in  1  square  chain  ?     In  4  ?     In  6  ? 

4.  How  many  square   chains   in   2  acres  ?     How   many 
perches  in  3  acres  ?     In5?     In  6? 


EXAMPLES. 


1.  How  many  perches  in 
32Jf.  25A  35.  19P.? 


OPERATION. 


323f.  25A  3P.  19P. 
640 


20505  acres. 
4 


82023  roods. 
40 


2.  How     many     square 


miles,  &c.,  in  3280989P.1 


OPERATION. 


40)3280939  19P. 

4)82023     37?. 
640)20505  25A 
32 

,     Ans.  321T.  25  A  ZR  19P. 

3280939  perches. 

3.  In  19A  272.  37P.,  how  many  square  rods  ? 

4.  In  175  square  chains,  how  many  square  feet  ? 

5.  In  37456  square  inches,  how  many  square  feet  ? 

6.  In  14972  perches,  how  many  acres  ? 

7.  In  3674139  perches,  how  many  square  miles? 

8.  Mr.  Wilson's  farm  contains   104A  3P.  and  19P.  ;  he 
paid  for  it  at  the  rate  of  75  cents  a  perch  :  what  did  it  cost? 

9.  The  four  walls  of  a  room  are  each  25  feet  in  length  and 
9  feet  in  height  and  the   ceiling  is  25  feet  square  :  how  much 
will  it  cost  to  plaster  it  at  9  cents  a  square  yard  ? 

CUBIC  MEASURE. 

110.  Cubic  measure  is  used  for  measuring  stone,  timber, 
earth,  and  such  other  things  as  have  the  three  dimensions, 
length,  breadth,  and  thickness. 

TABLE. 

1728  cubic  inches,  Cu.  in.,  make  1  cubic  foot,         Cu.  ft. 
27  cubic  feet,    -  1  cubic  yard,        Cu.  yd. 

40  feet  of  round  or         )  -,    .  n, 

50  feet  of  hewn  timber,  J 

42  cubic  feet,    -  1  ton  of  shipping,  T. 

16  cubic  feet,    -  -         1  cord  foot,          C.ft. 

8  cord  feet,  or )  .         ,  r 

128  cubic  feet,     \      '  l  cord' 


DENOMINATE  NUMBERS.  105 

NOTE.— 1.  A  cord  of  wood  is  a  pile  4  feet  wide,  4  feet  high, 
and  8  feet  long. 

2.  A  cord  foot  is  1  foot  in  length  of  the  pile  which  makes  a 
cord. 

3.  A  CUBE   is   a   figure   bounded  by  six   equal   squares,  called 
faces;  the  sides  of  the  squares  are  called  edges. 

4.  A  cubic  foot  is  a  cube,  each  of  whose  faces  is  a  square  foot, 
its  edges  are  each  1  foot. 

5.  A  cubic  yard  is  a  cube,  each  of 
whose  edges  is  1  yard. 

6.  The  base  of  a  cube  is  the  face 
on  which  it  stands     If  the  edge  of 
the  cube  is  one  yard,  it  will  contain 

3x3=9    square    feet ;     therefore,   9      ^  __ 

cubic  feet  can  be  placed  on  the  base,      j£ 

and  hence,  if  the  figure  were  1  foot 

thick,  it  would  contain  9  cubic  feet ;  d  feet-1 

if  it  were  2  feet  thick  it  would  contain  2  tiers  of  cubes,  or  18  cubic 

feet ;  if  it  were  3  feet  thick,  it  would  contain  27  cubic  feet ;  hence, 

The  contents  of  a  figure  of  this  form  are  found  by  multi- 
plying the  length,  breadth,  and  thickness  together. 

7.  A  ton  of  round  timber,  when  square,  is  supposed  to  produce 
40  cubic  feet ;  hence,  one-fifth  is  lost  by  squaring. 

1.  In  1  cubic  foot,  how  many  cubic  inches?     How  many 
in  2  ?     In  3  ? 

2.  In  1  cubic  yard,  how  many  cubic  feet  ?     How  many  in 
2  ?     In  4  ?     In  6  ? 

3.  How  many  cord  feet  in  3  cords  of  wood  ?  In  5  ?  In  6  ? 

4.  How  many  cubic  feet  in  2  cords  ?     In  half  a  cord,  how 
many  ?     How  many  in  a  quarter  of  a  cord  ? 

5.  How  many  cubic  yards  in  54  cubic  feet  ?     In  81  ? 

6.  In  120  feet  of  round^ timber,  how  many  tons  ? 

7.  How  many  tons  of  shipping  in  84  cubic  feet  ?    In  168  ? 

8.  How  many  cords  of  wood  in  64  cord  feet  ?     In  96  ?    In 
128? 

9.  How  many  cubic  feet  in  a  stone  8  feet  long,  3  feet 
wide  and  2  feet  thick  ? 

110.  For  what  is  cubic  measure  used  ?  What  are  its  denominations  ? 
What  is  a  cord  of  wood  ?  What  is  a  cord  foot  ?  What  is  a  cube  ? 
What  is  a  cubic  foot  ?  What  is  a  cubic  yard  ?  How  many  cubic  feet 
in  a  cubic  yard?  What  are  the  contents  of  a  solid  equal  to?  Repeat 
the  table.  What  are  the  units  of  the  scale  ? 


106  REDUCTION    OF 


EXAMPLES. 


1.  In  15cw.  yd.  IScu.  ft. 
16cw.  in.,  how  many  cubic 
inches  ? 


OPERATION. 


cu.  yd.  cu.  ft.    cu.  in. 
15         18  16 


113 

31 

423x1728  +  16=730960. 


2.  In  730960  cubic  inch- 
es, how  many  cubic  yards, 
&c.? 

OPERATION. 

1728)730960  cu.  in. 


27^423  cu.  ft.  16 
15ctt.yd.18 


cu.  yd.       cu.ft.  cu.  in. 
Ans.     15  18  16 


3.  How  many  small  blocks  1  inch  on  each  edge  can  be 
sawed  out  of  a  cube  7  feet  on  each  edge,  allowing  no  waste 
for  sawing  ? 

4.  In  25  cords  of  wood,  how  many  cord  feet  ?     How  many 
cubic  feet  ? 

5.  How  many  cords  of  wood  in  a  pile  28  feet  long,  4  feet 
wide,  and  6  feet  in  height  ? 

6.  In  174964  cord  feet,  how  many  cords? 

7.  In   7645900   cubic  inches,   how  many  tons  of  hewn 
timber  ? 

WINE  OR  LIQUID  MEASURE. 
111.  Wine  measure  is  used  for  measuring  all  liquids. 

TABLE. 


4  gills,  gi.             make 

1  pint,       marked 

pt. 

2  pints 

1  quart,     - 

qt. 

4  quarts 

1  gallon,    - 

gal. 

31  1  gallons    - 

1  barrel,    -          bar. 

or  bbl. 

42  gallons    - 

1  tierce 

tier, 

63  gallons     - 

1  hogshead, 

hhd. 

2  hogsheads 

1  pipe 

pi. 

2  pipes  or  4  hogsheads 

1  tun, 

tun. 

111.  What  is  measured  by  wine  or  liquid  measure  ?  What  are  its 
denominations  ?  Repeat  the  table.  What  are  the  units  of  the  scale  ? 
What  is  the  standard  wine  gallon? 


DENOMINATE  NUMBERS.  107 

gi.         pt.        qt.  gal.      bar.    tier.  hhd.  pi.  tun. 
4         =  1 

8         =2  =1 

32       =8  =4  =1 

1008  =252  =126  =311     -i 

1344   =336  =168  =42  =1 

2016  =504  =252  =63  =1$  =1 

4032   =1008  =504  =126  =3     =2  =  1 

8064  =2016  =1008  =252  =6     =4  =  2  =1 

NOTE. — The  standard  unit,  or  gallon  of  liquid  measure,  in  the 
United  States,  contains  231  cubic  inches. 

1.  How  many  gills  in  4  pints  ?     How  many  pints  in  3 
quarts  ?     In  6  quarts  ?     In  9  ?     In  10  ? 

2.  How  many  quarts  in  2  gallons  ?     In  4  gallons  ?     In  6 
gallons  ?     How  many  pints  in  2  gallons  ?     In  5  ? 

3.  How  many  barrels  in  a  hogshead  ?     How  many  in  4 
hogsheads  ?     In  6  ? 

4.  How  many  quarts  in  3  gallons?     In  5  gallons?    In  20? 
In  a  barrel  how  many  ?     In  a  hogshead  how  many  ? 


EXAMPLES. 


1.  In  5  tuns  3  hogsheads 
17  gallons  of  wine,  how 
many  gallons? 


OPERATION. 


btuns  3hhd.  17 gal. 
4 


23 
63 

76 


139 


2.  In  1466  gallons,  how- 
many  tuns,  &c.  ? 


OPERATION. 

63)1466 

4)23       17  gal. 
5         3  hhd. 


Ans.  Stuns  Bhhd.  llgal. 
14  66  gallons. 

3.  In  12  pipes  1  hogshead  and  1  quart  of  wine,  how  many 
pints  ? 

4.  In  10584  quarts  of  wine,  how  many  tuns  ? 

5.  In  201632  gills,  how  many  tuns? 

6    What  will  be  the  cost  of  3  hogsheads,  1  barrel,  8  gal- 
lons, and  2  quarts  of  vinegar,  at  4  cents  a  quart  ? 


108 


REDUCTION   OF 


ALE  OR  BEER  MEASURE. 

112.  Ale  or  Beer  Measure  was  formerly  used  for  mea- 
suring ale,  beer,  and  milk. 

TABLE, 
make  1  quart,       marked  qt. 

-  1  gallon,    - 

-  1  barrel,    - 

-  1  hogshead, 


2  pints,  pt. 

4  quarts 
36  gallons 
54  gallons 
pt. 


2 
8 

288 
432 


4 
144 

216 


gal. 


bar. 


gal. 
bar. 
hhd. 
hhd. 


=    1 

=  36         =1 
=  54         =11         =1 
NOTE. — 1  gallon,  ale  measure,  contains  282  cubic  inches. 

1.  How  many  pints  in  3  quarts  ?     How  many  in  5? 

2.  How  many  quarts  in  3  gallons  ?     In  4  gallons  ?     In  9  ? 

EXAMPLES. 


1  .  How  many  quarts  are 
there  in  ±hhd.  26ar. 


OPERATION. 

4hhd.  26ar.  Wgal.  8qt. 

li 

4 

4 

86ar. 
36 

57 
26 

317  gal. 


2.    In   1271    quarts,   how 
many  hogsheads,  &c.  ? 

OPERATION. 

4)1271 
36)317     Zqt. 


Ans.  ±hhd.  26ar.  ZSgal.  Zqt. 


3.  In^476ar.  Ifigal.  &qt.,  how  many  pints  ? 

4.  In  27Md.  36ar.  25</a/.  3(?£.,  how  many  pints  ? 

5.  In  55832  pints,  how  many  hogsheads  ? 

6.  In  64972  quarts,  how  many  barrels  ? 


112.  For  what  is  ale  or  beer  measure  used? 
iuatious  ?    Repeat  tho  table. 


What  are  its  denotn- 


DENOMINATE   NUMBERS. 


109 


DRY  MEASURE. 

113.  Dry  Measure  is  used  in  measuring  all  dry  articles, 
such  as  grain,  fruit,  salt,  coal,  &c. 


TABLE. 


2  pints,  pt. 

8  quarts   - 

4  pecks     - 

36  bushels  - 


make     1  quart,    marked 


1  peck,   - 


bushel, 
chaldron, 


bu. 


pk. 
bu. 
ch. 

ch. 


_ 


16  =8  =1 

64  =32  =4 

2304       =  1152       =  144 

1.  How  maty  quarts  in  2  pecks  ? 

2.  How  many  pecks  in  24  quarts  ? 

3.  How  many  pecks  in  6  bushels  ? 
many  bushels  in  16  pecks  ?     In  32  ? 


=  1 

=  36     =  1. 

In  5  ?     In  8  ? 
In  32  ?     In  64  ? 
In  8?     In  12?     How 
In  40? 


4.  How  many  bushels  in  2  chaldrons  ?     In  3  ?     In  4  ? 

NOTE.— The  standard  bushel  of  the  United  States  is  the  Win- 
chester bushel  of  England.  It  is  a  circular  measure,  18£  inches  in 
diameter  and  8  inches  deep,  and  contains  2150s  cubic  inches,  nearly. 

2.  A  gallon,  dry  measure,  contains  268£  cubic  inches. 


EXAMPLES. 


1.  How  many  quarts  are 
there  in   65c7i.  206w.  3pk. 

Iqt.  ?             OPERATION. 

£5c/i.  206w.  Bpk.  *lqt. 
36 

2.  How  many  chaldrons, 
&c.,  in  75551  quarts? 

OPERATION. 

8)75551 

390 
19T 

4)9443       Iqt. 
36)2360       Zplc. 

2360 
4 

65       206w. 

9443 

8 

Ans  65cA  20ta  3  Ic  7  t 

75551  quarts. 

113.  What  articles  are  measured  by  dry  measure?  What  are  its 
denominations?  Repeat  the  table.  What  J3  the  standard  bushel? 
What  arc  the  contents  of  a  gallon? 


110  REDUCTION   OF 

3.  In  312  bushels,  how  many  pints  ? 

4.  In  5  chaldrons  31  bushels,  how  many  pecks  ? 

5.  In  17408  pints,  how  many  bushels? 

6.  In  4220  pints,  how  many  chaldrons  ? 

AVOIRDUPOIS  WEIGHT. 

114.  By  this  weight  all  coarse  articles  are  weighed,  such 
as  hay,  grain,  chandlers'  wares,  and  all  metals  except  gold 
and  silver. 

TABLE. 

16  drams,  dr.    make  1  ounce,     marked  oz. 

16  ounces  1  pound,  lb. 

25  pounds  1  quarter,        -  qr. 

4  quarters  -         -  1  hundred  weight,  cwt. 

20  hundred  weight  1  ton,      -  T. 

qr.  cwt.  T. 


dr. 

oz. 

lb. 

16 

=  1 

256 

=  16 

=  1 

6400 

=  400 

=  25 

25600 

=  1600 

=  100 

=  4       =1 

512000     =  32000     =  2000     =  80     =  20         =  1 

NOTES. — 1.  The  standard  avoirdupois  pound   is  the  weight   of 
27.7015  cubic  inches  of  distilled  water. 

2.  By  the  old  method  of  weighing,  adopted  from  the  English 
system,  112  pounds  were  reckoned  for  a  hundred  weight.     But  now, 
the  laws  of  most  of  the  States,  as  well  as  general  usage,  fix  the 
hundred  weight  at  100  pounds. 

3.  The  units  of  the  scale,  in  passing  from  drams  to  ounces,  are 
16 ;    from  ounces  to  pounds,   16  ;    from  pounds  to  quarters,  25 ; 
from  quarters  to  hundreds,  4 ;  and  from  hundreds  to  tons,  20. 

1.  In  2oz.,  how  many  drams  ?     In  3  ?     In  4  ?     In  5. 

2.  In  4/6.,  how  many  ounces  ?     In  3  how  many  ?     In  2  ? 

3.  In  Qqr.,  how  many  hundred  weight  ?     In  bqr.  ? 

4.  In  Scwt.,  how  many  quarters  ?     How  many  in  ±cwt.  ? 

5.  In  60  hundred  weight,  how  many  tons  ?     In  80  ? 


114.  For  what  is  avoirdupois  weight  used  ?  How  is  the  table  to  be 
read  ?  How  can  you  determine,  from  the  second  table,  the  value  of 
any  unit  in  units  of  the  lower  denominations  ? 


DENOMINATE   NUMBERS. 


Ill 


EXAMPLES. 


1.  How  many  pounds  are 
there    in    15T.    Scurf.    3qr.  ' 
15/6.  ? 

OPERATION. 

15  T.  Scwt.  Sqr.  15/6. 
20 


308  cwtt 
4 


1235  qr. 

25 

~6l80     5  /6.  added. 
2471       1  ten  added. 
30890  Ib. 


2.  In  30890  pounds,  how 
many  tons  ? 

OPERATION. 

25)30890 
4)1235gr.      15/6. 
.     Zqr. 
Scwt. 


Ans.  15  T.  8cut.  3qr.  15/6. 


3.  In  5T.  Scurf.  3#r.  24/6.  13oz.  14dr.,  how  many  drams  ? 

4.  In  28 T.  4curf.  Iqr.  21/6.,  how  many  ounces? 

5.  In  2790366  drams,  how  many  tons? 

6.  In  903136  ounces,  how  many  tons? 

7.  In  3124446  drams,  how  many  tons? 

8.  In  93 T.  13cwrf.  3qr.  8/6.,  how  many  ounces? 

9.  In  108910592  drams,  how  many  tons  ? 

10.  What  will  be  the  cost  of  11  T.  17curf.  Sqr.  24/6.  of  hay 
at  half  a  cent  a  pound  ?     How  much  would  that  be  a  ton  ? 

11.  What  is  the  cost  of  2T.  13cw;/.  3?r.  21/6.  of  beef  at 
8  cents  a  pound  ?     How  much  would  that  be  a  ton  ? 

TROT  WEIGHT. 

115.  Gold,   silver,  jewels,    and  liquors,   are  weighed   by 
Troy  weight. 

TABLE. 

24  grains,  gr.         make  1  pennyweight,  marked  pwt. 
20  pennyweights     -         1  ounce     -        -        -    oz. 
12  ounces       -  1  pound    -  -    Ib. 


gr. 
24 

480 
5760 


pwt. 
=  1 
=  20 
=  240 


oz. 

i 

=  12 


Ib. 


—  1. 


112  REDUCTION   OF 

NOTES.— 1.  The  standard  Troy  pound  is  the  weight  of  22.794377 
cubic  inches  of  distilled  water.  It  is  less  than  the  pound  avoirdupois. 

2.  The  units  of  the  scale,  in  passing  from  grains  to  penny- 
weights, are  24 ;  from  pennyweights  to  ounces,  20 ;  and  from 
ounces  to  pounds,  12. 

1.  How  many  grains  in  2  pennyweights  ?     In  3  ?     In  4  ? 

2.  How  many  pennyweights  in  48  grains  ?     In  72  ? 

3.  How  many  ounces  in  40  pennyweights  ?     In  60  ? 

4.  How  many  ounces  in  4  pounds  ?    In  12  ?    In  9?    In  7  ? 

5.  How  many  pounds  in  24  ounces  ?     In  36  ?"  In  96  ? 


EXAMPLES. 


1.   How  many  grains  are 
there  in    16/6.   lloz.  Ibpwt. 


OPERATION. 

16/6.  lloz.  15pwt.  17or. 
12 

203  ounces. 
20 

4075  pennyweights 
24 


97817  grains. 


2.  In   97817   grains,  how 
many  pounds  ? 


OPERATION. 


24)97817 

20)4075  pwt.  17or. 

12)  203  oz.  15pwt. 

16/6.  lloz. 


Ans.  16/6.  lloz.  I5pwt.  11  gr. 


3.  In  25/6.  9oz.  20or.,  how  many  grains  ? 

4.  In  6490  grains,  how  many  pounds  ? 

5.  In  148340  grains,  how  many  pounds  ? 

6.  In  117/6.  9oz.  Ibpwt.  ISgr.,  how  many  grains  ? 

7.  In  8794pio/.,  how  many  pounds  ? 

8.  In  6/6.  9oz.  21grr.,  how  many  grains  ? 

9.  In  1/6.  loz.  Wpivt.  16#?\,  how  many  grains  t 

10.  A  jewel  weighing  2oz.  \±pwt.  18</r.,  is  sold  for  half  a 
dollar  a  grain  :  what  is  its  value  ? 

Notes.  1. — What  is  the  standard  avoirdupois  pound  ? 

2.— What  is  a  hundred  weight  by  the  English  method?  What  is  a 
hundred  weight  by  the  United  States  method  ? 

'.>.  Name  the  units  of  the  scale  in  passing  from  one  denomination  to 
another. 

115.  What  articles  are  weighed  bv  Troy  weight  ?  What  arc  its  de- 
nominations? Repeat  the  table?  What  is  the  standard  Troy  pound  ? 
What  arc  the  units  of  the  scale,  in  passing  from  one  unit  to  another  ? 


DENOMINATE   NUMBERS. 


113 


APOTHECARIES'  WEIGHT. 

110.  This  weight  is  used  by  apothecaries  and  physicians 
in  mixing  their  medicines.  But  medicines  are  generally  sold, 
in  the  quantity,  by  avoirdupois  weight 

TABLE. 

20  grains,  gr.  make  1  scruple,  marked  3. 

3  scruples  -  -  1  dram,  -  -  -  3  • 

8  drams  -  -  -  1  ounce,  -  -  -  | . 

12  ounces-    -    -       1  pound,   -    -    -  fi>. 


gr. 

20 
60 

480 
5760 


3 
1 

3 
24 

288 


.1 
8 
96 


I 


__  •» 

=  12 


=  1 


NOTES. — 1.  The  pound  and  ounces  are  the  same  as  the  pound 
and  ounce  in  Troy  weight. 

2.  The  units  of  the  scale,  in  passing  from  grains  to  scruples, 
are  20 ;  in  passing  from  scruples  to  drams,  3 ;  from  drams  to 
ounces,  8 ;  and  from  ounces  to  pounds,  12. 

1.  How  many  grains  in  2  scruples  ?  In  3  ?    In  4  ?    In  6  ? 

2.  How  many  scruples  in  4  drams  ?  In  7  drams  ?     In  5  ? 

3.  How  many  drams  hi  5  ounces  ?  How  many  ounces  in 
32  drams  ? 


EXAMPLES. 


1.  How  many  grains  in 
>  8  §    63    23 


OPERATION. 

9fi>  8  3    63 
12 


23 


116  ounces. 
8 

9d4  scruples. 
_3 

2804  drams. 
20 

56092  grains. 


2.  In   56092  grains,  how 
many  pounds  ? 


OPERATION. 

20)56092 

3)28043 
~8)9343 


23 

63 

81 


Am.  9fi>  8  |   63  23 


REDUCTION   OF 


3.  In  27  ft>  9§   63   13,  bow  many  scruples  ? 

4.  In  94ft)  11  |    13,  how  many  drams  ? 

5.  8011  scruples,  how  many  pounds? 

6.  In  9113  drams,  how  many  pounds  ? 

7.  How  many  grains  in  12ft>  9  §   73  23 

8.  In  73918  grains,  how  many  pounds? 


MEASURE  OF  TIME. 

117.  TIME  is  a  part  of  duration.  The  time  in  which  the 
earth  revolves  on  its  axis  is  called  a  day.  The  time  in  which 
it  goes  round  the  sun  is  365  days  and  6  hours,  and  is  called  a 
year.  Time  is  divided  into  parts  according  to  the  following 


TABLE. 


60  seconds,  sec. 
60  minutes  - 
24  hours     - 

7  days 

4  weeks    - 
13 wo.  Ida.  and  6/irs.; 

or  365  da.  Qhr. 
12  calendar  months    - 


sec. 
60 

3600 
86400 
604800 


m. 
=  1 

=  60 
—  1440 

=  10080 


nak 

e  1  minute,         marked 
1  hour, 
1  day, 
1  week, 
1  month, 

m. 
hr. 
da. 
wk. 
mo. 

j- 

1  Julian  year, 

yr. 

- 

1  year, 

yr. 

hr.            da.             wk. 

__ 

1 

•  — 

24         =  1 

— 

168       =7           =1 

yr. 


31557600     =  525960     =  8766     =  365J     =52     =1 

NOTES. — 1.  The  years  are  numbered  from  the  beginning  of  the 
Christian  Era.  The  year  is  divided  into  12  calendar  months, 
numbered  from  January :  the  dtays  are  numbered  from  the  begin- 
ning of  the  month  :  hours  from  12  at  night  and  12  at  noon. 


.  31 

-  31 
.  30 

-  31 

.  30 

.  31 


Names. 
January,-     - 
February,     - 
March,     -    - 
April,  -    -    - 
May,    -    .    . 
June    -    -    - 

No. 
-    1st. 
-    2d. 
-    3d. 
-    4th. 
-    5th. 
.    6th. 

No.  i 

lays. 
31 
28 
31 
30 
01 
30 

Names. 
July,     -    -    - 
August,    -    - 
September,   - 
October,    -    - 
November,    - 
December,     - 

No. 
.    7th. 
-    8th. 
-    9th. 
-  10th. 
-  llth. 
-  12th. 

DENOMINATE   NUMBERS.  115 

2.  The  leogth  of  the  tropical  year  is   365<J.  57ir.  48m.  4Ssec. 
nearly  ;  but  in  the  examples  we  shall  regard  it  as  365d.  6/w. 

3.  Since  the  length  of  the  year  is  365  days  and  6  hours,  the  odd 
G  hours,  by  accumulating  for  4  years,  make  1  day,  so  that  every 
fourth  year  contains  366  days.     This  is  called  Bissextile  or  Leap 
Year.     The  leap  years  are  exactly  divisible  by  4:  1872,  1876,  1880, 
are  leap  years. 

4.  The  additional  day,  when  it  occurs,  is  added  to  the  month  of 
February,  BO  that  this  month  has  29  days  in  the  leap  year. 

Thirty  days  hath  September, 
April,  June,  and  November ; 
All  the  rest  have  thirty-one, 
Excepting  February,  twenty-eight  alone. 

1.  How  many  seconds  in  4  minutes  ?     How  many  in  6  ? 

2.  How  many  hours  in  3  days  ?     How  many  in  5  ?     In  8  ? 

3.  How  many  days  in  6  weeks  ?     In  8,  how  many  ? 

4.  How  many  hours  in  1  week  ?  How  many  weeks  in  42da.  ? 


EXAMPLES. 


1.  How  many  seconds  in 
Qhr.  ? 


OPERATION. 


365da.  6/ir. 
24 

1466 
730 


2.  How  many  days,  &c. 
in  31557600  seconds? 

OPERATION. 

60)31557600 
60)525960 


24)8766 


365       6/ir. 
Ans.  365tfa.  Qhr. 


8766 
60 

525960x60  =  31557600  sec. 

3.  If  the  length  of  the  year  were  365da.  23/ir.  57m.  39sec., 
how  many  seconds  would  there  be  in  12  years? 

4.  In  126230400  seconds,  how  many  years  of  365  days? 

5.  In  756952018  seconds,  how  many  years  of  365  days  ? 

117.  What  are  the  denominations  of  time?  How  long  is  a  year? 
How  many  days  in  a  common  year?  How  many  days  in  a  Leap  year? 
How  many  calendar  months  in  a  year  ?  Name  them,  and  the  number 
of  days  in  each.  How  many  days  has  Februarv  in  the  leap  year  ?  How 
do  you  remember  which  of  the  months  have  30  days,  and  which  31  ? 


116  REDUCTION   OF 

6.  In  285290205  seconds,  how  many  years  of  365da.  6Ar. 
each? 

7.  How  many  hours  in  any  year  from  the  31st  day  of  March 
to  the  1st  day  of  January  following,  neither  day  named  being 
counted  ? 

CIRCULAE  MEASURE. 

118.  Circular  measure  is  used  in  estimating  latitude  and 
longitude,  and  also  in  measuring  the  motions  of  the  heavenly 
bodies. 

The  circumference  of  every  circle  is  supposed  to  be  divided 
into  360  equal  parts,  called  degrees.  Each  degree  is  divided 
into  60  minutes,  and  each  minute  into  60  seconds. 

TABLE. 

60  seconds'         make  1  minute,     marked  '. 

60  minutes  1  degree,  -        -       °. 

30  degrees  -  1  sign  s. 

12  signs  or  360°         -  1  circle  c. 


60  =  1 

3600  =60  =1 

108000  =  1800  =30  =1 

1296000  =  21600  =360  =12         =1 

1.  How  many  seconds  in  3  minutes  ?     In  4  ?     In  5  ? 

2.  How  many  minutes  in  6  degrees  ?     In  4  ?     In  5  ? 

3.  How  many  degrees  in  4  signs  ?     In  6  ?     In  7  ?     In  8  ? 

4.  How  many  degrees  in  240  minutes  ?     In  720  ?     How 
many  signs  in  90°  ?     In  150°  ?     In  180°  ? 

EXAMPLES. 

1.  In  5s.  29°  25',  how  many  minutes  ? 

2.  In  2  circles,  how  many  seconds  ? 

3.  In  27894  seconds,  how  many  degrees,  &c.  ? 

4.  In  32295  minutes,  how  many  circles,  &c.  ? 

5.  In  3  circles  16°  20',  how  many  seconds  : 

6.  In  8s.  16°  25",  how  many  seconds  ? 

7.  In  8589  seconds,  how  many  degrees,  &c.  ? 

118.  For  what  is  circular  measure  used?    How  is  every  circle  sup- 
posed to  be  divided  ?    Repeat  the  table. 


DENOMINATE   NUMBERS.  117 

MISCELLANEOUS  TABLE. 

12  units,  or  things  make     1  dozen. 

12  dozen     -  1  gross. 

12  gross,  or  144  dozen  '               1  great  gross. 

20  things    -  1  score. 

100  pounds  -  1  quintal  of  fish. 

196  pounds  1  barrel  of  flour. 

200  pounds  -  1  barrel  of  pork. 

18  inches  1  cubit. 

22  inches,  nearly  -  1  sacred  cubit. 

14  pounds  of  iron  or  lead  -         1  stone. 

21 J  stones     -  1  pig. 

8  pigs  1  fother. 

BOOKS  AND  PAPER. 

The  terms,  folio,  quarto,  octavo,  duodecimo,  &c.,  indicate 
the  number  of  leaves  into  which  a  sheet  of  paper  is  folded. 

A  sheet  folded  in    2  leaves  is  called  a  folio. 

A  sheet  folded  in    4  leaves  "  a  quarto,  or  4to. 

A  sheet  folded  in    8  leaves  "  an  octavo,  or  8vo 

A  sheet  folded  in  12  leaves  "  a  12mo. 

A  sheet  folded  in  16  leaves  "  a  16mo. 

A  sheet  folded  in  18  leaves  "  an  18mo 

A  sheet  folded  in  24  leaves  "  a  24mo. 

A  sheet  folded  in  32  leaves  "  a  32mo. 

24  sheets  of  paper  make             1  quire. 

20  quires  -  -                  1  ream. 

2  reams  -  1  bundle. 

5  bundles  1  bale. 

MISCELLANEOUS    EXAMPLES. 

1.  How  many  hours  in  344wfc.  Qda.  llhr.  ? 

2.  In  6  signs,  how  many  minutes  ? 

3.  In  15  tons  of  hewn  timber,  how  many  cubic  inches  ?• 

4.  In  171360  pence,  how  many  pounds? 

5.  In  1720320  drams,  how  many  tons? 

6.  In  55799  grains  of  laudanum,  how  many  pounds? 

7.  In  9739  grains,  how  many  pounds  Troy? 

8.  In  59/6.  ISpwt.  5grr.,  how  many  grains  ? 

9.  In  .£85  8s.,  how  many  guineas  ? 

10.  In  346  E.  F.,  how  many  Ells  English  ? 

i 


118  KEDUCTION   OF 

11.  In  3hhd.  ISgal.  2qt.,  how  many  half-pints  ? 

12.  In  12 T.  Ibcwt.  Iqr.  1Mb.  12dr.,  how  many  drams? 

13.  In  40144896  square  inches,  how  many  acres? 

14.  In  5760  grains  Troy,  how  many  pounds? 

15.  In  6  years  (of  52  weeks  each),  3>2wk.  bda.  17/ir.,  how 
many  hours  ? 

16.  In  811480",  how  many  signs  ? 

17.  In  2654208  cubic  inches,  how  many  cords  ? 

18.  In  18  tons  of  round  timber,  how  many  cubic  inches  ? 

19.  In  84  chaldrons  of  coal,  how  many  pecks? 

20.  In  302  ells  English,  how  many  yards  ? 

21.  In  Qihhd.  ISgal.  2qt.  of  molasses,  how  many  gills  ? 

22.  In  76 A  IB.  8P.,  how  many  square  inches? 

23.  In  £15  19s.  lid.  3/ar.,  how  many  farthings? 

24.  In  445577  feet,  how  many  miles? 

25.  In  37444325  square  inches,  how  many  acres  ? 

26.  If  the  entire  surface  of  the  earth  is  found  to  contain 
791300159907840000  square  inches,  how  many  square  miles 
are  there  ? 

27.  How  many  times  will  a  wheel  16  feet  and  6  inches  in 
circumference,  turn  round  in  a  distance  of  84  miles  ? 

28.  What  will  28  rods,  129  square  feet  of  land  cost  at  $12 
a  square  foot  ? 

29.  What  will  be  the  cost  of  a  pile  of  wood  36  feet  long 
6  feet  high  and  4  feet  wide,  at  50  cents  a  cord  foot  ? 

30.  A  man  has  a  journey  to  perform  of  288  miles.     He 
travels  the  distance  in  12  days,  travelling  6  hours  each  day : 
at  what  rate  does  he  travel  per  hour  ? 

31.  How  many  yards  of  carpeting  1  yard  wide,  will  carpet 
a  room  18  feet  by  20? 

32.  If  the  number  of  inhabitants  in  the  United  States  is 
24  millions,  how  long  will  it  take  a  person  to  count  them, 
counting  at  the  rate  of  100  a  minute  ? 

33.  A  merchant  wishes  to  bottle  a  cask  of  wine  containing 
126  gallons,  in  bottles  containing  1  pint  each  :  how  many 
bottles  are  necessary  ? 

34.  There  is  a  cube,  or  square  piece  of  wood,  4  feet  each 
way :  how  many  small  cubes  of  1   inch  each  way,  can  be 
sawed  from  it,  allowing  no  waste  in  sawing  ? 

35.  A  merchant  wishes  to  ship  285  bushels  of  flax-seed  in 
casks  containing  7  bushels  2  pecks  each :  what  number  of 
casks  are  required  ? 


DENOMINATE   NUMBERS  119 

36.  How  many  times  will  the  wheel  of  a  car,  10  feet  and 
6  inches  in  circumference,  turn  round  in  going  from  Hartford 
to  New  Haven,  a  distance  of  34  miles  ? 

37.  How  many  seconds  old  is  a  man  who  has  lived  32 
years  and  40  days  ? 

38.  There  are  15713280  inches  in  the  distance  from  New 
York  to  Boston,  how  many  miles  ? 

39.  What  will  be  the  cost  of  3  loads  of  hay,  each  weighing 
IScwt.  3qr.  24/6.,  at  7  mills  a  pound? 

ADDITION  OF  DENOMINATE  NUMBERS. 

119.  Addition  of  denominate  numbers  is  the  operation  of 
finding  a  single  number  equivalent  in  value  to  two  or  more 
given  numbers.  Such  single  number  is  called  the  sum. 

How  many  pounds,  shillings,  and  pence  in  £4  8s.  9c?., 
£27  14s.  lid.,  and  £156  17s.  lOd.  ? 

ANALYSIS. — We  write  the  units  of  the  same  OPERATION. 

name  in  the  same  column.     Add  the  column  £.     s.     d. 

of  pence ;    then  30  pence  are  equal  to  2  shil-  489 

lings  and  6  pence  :  writing  down  the  6,  carrying  9*   -.  -    , , 

the  two  to  the  shillings.     Find  the  sum  of  the  JJ    1J    iL 
shillings,  which  is  41 ;  that  is,  2  pounds  and  1 

shilling  over.     Write  down  1*. ;  then,  carrying  ^£189      ls<   g^ 
the  2  to  the   column   of  pounds,  we  find  the 
sum  to  be  £189  Is.  6d. 

NOTE. — In  simple  numbers,  the  number  of  units  of  the  scale, 
at  any  place,  is  always  10.  Hence,  we  carry  1  for  every  10.  In 
denominate  numbers,  the  scale  varies.  The  number  of  units,  in 
passing  from  pence  to  shillings,  is  12  ;  hence,  we  carry  one  for 
every  12.  In  passing  from  shillings  to  pounds,  it  is  20  ;  hence,  we 
carry  one  for  every  20.  In  passing  from  one  denomination  to 
another,  we  carry  1  for  so  many  units  as  are  contained  in  the  scale 
at  that  place.  Hence,  for  the  addition  of  denominate  numbers,  we 
have  the  following 

RULE. — I.  Set  down  the  numbers  so  that  units  of  the 
same  name  shall  stand  in  the  same  column  ; 

II.  Add  as  in  simple  numbers,  and  carry  from  one  de- 
nomination to  another  according  to  the  scale. 
PROOF. — The  same  as  in  simple  numbers. 

119.  What  is  addition  of  denominate  numbers?  How  do  .you  set 
down  the  numbers  for  addition  ?  How  do  you  add  ?  How  do  you 
prove  addition  ?  ^- 


ADDITION  OF 


£  (8} 
173  13 

87  17 
75  18 

d. 
5 

7* 

EXAMPLES. 

(2.) 
£  s  d 
705  17  3J 
354  17  2j 
175  17  3| 

(3.; 

£  s. 
104  18 
404  17 
467  11 

I 
d. 
9| 

'4 

25 

17 

4 

87 

19  71 

597  14 

*i 

10 

10 

i°i 

52 

12  7| 

22  18 

5 

373 

18 

3 

18  6 

5 

TROY  WEIGHT. 

(4.) 

(5.) 

Ib. 

oz. 

pwt. 

gr. 

Ib. 

oz. 

pwt. 

gr. 

Ldd 

100 

10 

19 

20 

171 

6 

13 

14 

432 

6 

0 

5 

391 

11 

9 

12 

80 

3 

2 

1 

230 

6 

6 

13 

7 

0 

0 

9 

94 

7 

3 

18 

0 

11 

10 

23 

42 

10 

15 

20 

0 

0 

8 

9 

31 

0 

0 

21 

APOTHECARIES'  WEIGHT. 

(6.)  (7.)                         (8.) 

ft)        !     3    3    gr.  I      3    3    gr.            33    gr. 

24       7     2     1     16  11     2     1     17             3     2     15 

17     It     7     2     19  7     4     2     14            0     1     13 

36       6     5     0       7  4     0     1     19             2     2     11 

15       9     7     1     13  2     5     2     11             7     0     17 

93419  10     1     2     16             5     2     14 

AVOIRDUPOIS  WEIGHT. 

(9.)  (10.) 

cwt.  qr.    Ib.      oz.  dr.  T.     cwt.  qr.    Ib.     oz. 

14     2       0     14       9  15     12  1     10     10 

13     2     20       1     15  71       8  2       6       0 

93673  83     19  3     15       5 

10     0     18     12     11  36       7  0     20     14 

73232  47     11  2       2     11 

6     1     19       8       1  63       5  2     19       7 

4  ,  3       0     15       5  12     13  1     14       9 

12     2       0       0     13  9       7  0       5     10 


DENOMINATE   NUMBERS.  121 

11.  A  merchant  bought  4  barrels  of  potash  of  the  following 
weights,  viz. :  1st,  3cwt.  2qr.  Mb.  12oz.  3dr.  ;  2d,  ±cwt.  Iqr. 
21/6.  4oz.  ;  3d,  ±cwt ;  4th,  icwt.  Qqr.  2/6.  15oz.  15dr.  : 
what  was  the  entire  weight  of  the  four  barrels  ? 


LONG  MEASURE. 

L. 
16 

.<"•< 
mi.  fur. 

2     7 

i 
rd.  yd.  ft. 
39     9     2 

rd. 
16 

yd.  ft. 

9     2 

171. 
11 

327 

1 

2 

20     7     1 

12 

11 

1 

9 

87 

0 

1 

15     6     1 

18 

14 

0 

7 

1 

1 

1 

1     2     2 

19 

15 

2 

1 

CLOTH  MEASURE. 

(14.) 
E.  Fl  qr. 
126     4 

na. 
4 

(15.) 
yd.  qr. 
4     3 

na. 
2 

E.E. 

128 

(16.) 
qr.  na. 
5     1 

in. 
3 

65 

3 

1 

5     4 

1 

20 

3 

1 

2 

72 

1 

3 

6     1 

0 

19 

1 

4 

1 

157 

2 

3 

25     2 

2 

15 

3 

1 

2 

LAND  OR  SQUARE  MEASURE. 

(17.)  (18.) 

Sq.  yd.  Sq.ft.  Sq.  in.  M.  A.  R.  P.  Sq.yd 

97         4         104  2  60  3  37  25 

22         3           27  6  375  2  25  21 

105         8             2  7  450  1  31  20 

37         7         127  11  30  0  25  19 

19.  There  are  4  fields,  the  1st  contains  12A  2P.  38P.  ; 
the  2d,  4: A.  IR.  26P.  ;  the  3d,  85 A  QR.  19P.  ;  arid  the 
4th,  57  A  IR.  2P.  :  how  many  acres  in  the  four  fields  ? 

CUBIC  MEASURE. 

(20.)  (21.)         (22.) 

Cu.yd.  Cu.ft.  Cu.in.  C.   S.ft.  C.  Cord  ft. 

65    25   1129  16   127  87    9 

37    26    132  17    12  26    7 

50     1   1064  18   119  16    6 

22    19     17  37   104  19    5 


122  ADDITION   OF 

WINE  OR  LIQUID  MEASURE. 

(23.)  (24.) 

hhd.  gal.  qt.  pt.                 tun.  pi.  hhd.  gal.  qt. 

127     65     3     2                     14     2     1     27  3 

12     60     2     3                     15     1     2     25  2 

450     29     0     1                       4     2     1     27  1 

21       023                       501     62  3 

14     39     1     2                       7     1     2     21  2 


DRY  MEASURE. 

(25.)  (26.) 

ch.    bu.  pk.  qt.  pt.  ch.  bu.  pk.    qt.  pt. 

27     25     3     7     1  141  36     3     7     2 

59     21     2     6     3  21  32     2     4     1 

21271  85  9103 

5       9182  10  4413 

TIME. 

(27.)  (28.) 

yr.    mo.  wk.  da.  hr.  wk.  da.   hr.    m.    sec. 

*  4     11     3     6     20  8     8  14     55     57 

3     10     2     5     21  10     7  23     57     49 

5       8     1     4     19  20     6  14     42     01 

101       9     3     7     23  6     5  23     19     59 

55       8     4     6     17  2     2  20     45     48 


CIRCULAR  MEASURE  OR  MOTION. 

(29.)  (30.) 

s.     °        '       "  s.      °        '  " 

5     17     36     29  6     29     27  49 

7  25     41     21  8     18     29  16 

8  15     16     09  7     09     04  58 


NOTE. — Since  12  signs  make  a  circumference  of  a   circle,  we 
write  down  only  the  excess  over  exact  12's. 

APPLICATIONS    IN    ADDITION. 

1.  Add  46/6.  9oz.  Ifywot.  16<?r.,  87/6.  lOoz.  Gpwt.  Ugr., 
100/6.  lOoz  Wpwt.  10#r.,  and  56/6.  Zpwt.  6gr.  together. 


DENOMINATE  NUMBERS.  123 

2.  What  is  the  weight  of  forty-six  pounds,  eight  ounces, 
thirteen  pennyweights,  fourteen  grains  ;  ninety-seven  pounds, 
three  ounces  ;  and  one  hundred  pounds,  five  ounces,  ten  pen- 
nyweights and  thirteen  grains  ? 

3,  Add  the  following  together:    29  T.  Ibcwt.  Iqr.  14/6. 
12oz.  Mr.,  IScwt.  3?r,  lib.,   50  17.   3?r    4oz.,   and  2T.   Iqr. 


4    What  is  the  weight  of  39  T.  Wcwt.  2?r.  2/6.  15oz.  I2dr., 
llcwt.  6/6.,  I2cwt.  3?r.,  and  2?r.  Sib.  Mr.l 

5.  What  is  the  sum  of  the  following  :  314^4.  2E.  39P. 
200s7.  ft-   136s?.  in.,   UA.  IE.  20P.  10s?.  ft.,  BE.  36P. 
and  4  A.  IE.  16P.? 

6.  What  is  the  solid  content  of  64fons  33/2.  800m.,  Qtons 
1200m.,  25/35.,  700m.,  and  95tes  31/fc  1500m. 

7.  Add  together,  966u.  3p&.  2qt.  Ipt.,  466w.  3pfc.  1?£.  Ipt., 
2pk.  Iqt.  Ipt.  and  236w.  3p&.  4?£.  lp£. 

8.  What  is  the  area  of  the  four  following  pieces  of  land  ; 
the  first  containing  20  A.  BE.  15P.  250s?.  ft.  116s?.  in.  ;  the 
second,  19A  IE.  39P.  ;  the  third,  2P.  10P.  60s?,  ft.  ;  and 
the  fourth,  5  A.  6P.  50s?.  in.  ? 

9.  A  farmer  raised  from  one  field  37Zw.  Ipk.  3qt.  of  wheat  ; 
from  a  second,  416w.  2pk.  5?£.  of  barley  ;  from  a  third,  356w. 
Ipk.  3qt.  of  rye  ;  from  a  fourth,  436w.  3pk.  Iqt.  of  oats  ;  how 
much  grain  did  he  raise  in  all  ? 

10.  A  grocer  received  an  invoice  of  4hhd.  of  sugar  ;  the 
first  weighed  llcwt.  15/6.  ;  the  second,  12cwt.  3?r.  15/6.  ;  the 
third,  Scwt.  Iqr.  16/6.  ;  the  fourth,  I2cwt.  Iqr.  :  how  much 
did  the  four  weigh  ? 

11.  A  lady  purchased  32?/ds.  3?rs.  of  sheeting  ;  31yds.  Iqr. 
of  shirting  ;  llyds.  2?rs.  of  linen  ;  and  Qyds.  2??*s.  of  cambric  : 
what  was  the  whole  number  of  yards  purchased  ? 

12.  Purchased  a  silver  teapot  weighing  23oz.  llpivt.  llgr.  ; 
a  sugar  bowl,  weighing  8oz.  ISpwt.  l$gr.  •  a  cream  pitcher, 
weighing  5oz.  ll^r.  :  what  was  the  weight  of  the  whole  ? 

13.  A  stage  goes  one  day,  87m.  Qfur.  24rd.  ?  the  next,  75??i. 
3/wr.  17r^.  ;  the  third,  80m.  Ifur.  Wrd.  ;  the  fourth,  78m. 
5/*wr.  :  how  far  does  it  go  in  the  four  days  ? 

14.  Bought  three  pieces  of  land  ;  the  first  contained  17 
acres  IE.  35?'rf.  ;  the  second,  36  acres  2E.  2lrd.  ;  and  the 
third,  46  acres  QE.  37rd.  :  how  much  land  did  I  purchase  ? 


124:  SUBTRACTION   OF 


SUBTRACTION  OF  DENOMINATE  NUMBERS. 

120.  The  difference  between  two  denominate  numbers  is 
such  a  number  as  added  to  the  less  will  give  the  greater. 
SUBTRACTION  is  the  operation  of  finding  this  difference. 

I.  What  is  the  difference  between  £27  16s    Sd   and  £19 
17s.  9df.? 

ANALYSIS. — We  cannot  take  9rf.  from  Sd. ;          OPERATION. 
we  therefore  add  to  the  upper  number  as  many  20     12 

units  as  are  contained  in  the  scale,  and  at  the         x**™      IDS.   8a. 
same  time  add  1,  mentally,  to  the  next  higher  19      17      9 

denomination  of  the  subtrahend.  We  then  say,  »      To    rr~ 

9  from  20  leaves  11.  Then,  as  we  cannot  sub- 
tract 18  from  1C,'  we  add  20  and  say,  18  from  36  leaves  18.  Now, 
as  we  have  taken  1  pound=20  shillings,  from  the  pounds,  and 
added  it  to  the  shillings,  there  are  but  26  pounds  left.  We  may 
then  say,  19  from  26  leaves  7,  or  20  from  27  leaves  7.  The  lat- 
ter is  the  easiest  in  practice. 

The  first  step  is  called  borrowing,  the  second,  carrying  :  hence, 

RULE. — I.  Set  down  the  less  number  under  the  greater, 
placing  units  of  the  same  value  in  the  same  column. 

II.  Begin  with  the  lowest  denomination,  and  subtract  as  in 
simple  numbers,  borrowing  and  carrying  for  each  operation 
according  to  the  scale. 

PROOF. — The  same  as  in  simple  numbers. 
EXAMPLES. 

(1.)  (2-) 

A.    E.    P.  T.  cwt.  qr.   Ib. 

From       -     18     3     28  4     12     3     20 

Take       -     15     2     30 )  2     18 _  3       1) 

Remainder  ~3     (T~38  )  1     14     0 19  ) 

Proof      -     liTir~28  4     12     3     20 

(3.)  (4  ) 

Ib.      oz.  pwt.  gr.                  Ib.     oz.   pwt.  gr. 

From       -     273       0  0       0                   18       9     10       0 

Take       -       98     10  18     21                     9     10     15     20 
Remainder 


DENOMINATE   NUMBERS. 


125 


(5.) 

T,  cwt.  qr.  Ib.    oz. 

From      -     7     14     1  3       6 

Take       -     2       6     3  4     11 
Remainder 

T.     hhd.  gal.  qt.  pt. 

From      -       151       3  50     3     2 

Take       -         27       2  54     3     2 
Remainder 


(6.) 

cwt.  qr.    Ib.  oz.  dr. 

14     2     12  10  8 

6     3     16  15  3 


(8.) 

yr.  wk.  da.  hr.  ' 
95  25  4  20  45  50 
80  30  6  23  46  56 


TIME  BETWEEN  DATES. 
121.  To  find  the  time   between  any  two  dates. 

1.  What  time  elapsed  between  July  5th,  1848,  and  August 
8th.,  1850  ? 


OPERATION. 

yr.    mo.  da. 
1850     8     8 
1848     7     5 
213 


NOTE. — In  the  first  date,  the  number  of 
the  year  is  1848 ;  the  number  of  the  month 
7,  and  the  number  of  the  day,  5.  In  the 
second  date,  the  number  of  the  year  is  1850, 
the  number  of  the  month  8,  and  the  number 
of  the  day,  8. 

Hence,  to  find  the  time  between  two  dates : 

Write  the  numbers  of  the  earlier  date  under  those  of  the 
later,  and  subtract  according  to  the  preceding  rule. 

NOTE.— 1.  In  finding  the  difference  between  dates,  as  in  casting 
interest,  the  month  is  regarded  as  the  twelfth  part  of  a  year,  and 
as  containing  30  days. 

2.  The  civil  day  begins  and  ends  at  12  o'clock  at  night. 

2.  What  is  the   difference  of  time  between  March  2d, 
1847,  and  July  4th,  1856? 

3.  What  is  the  difference  of  time  between  April  28th,  1834, 
and  February  3d,  1856  ? 

4.  What  time  elapsed  between  November  29th,  1836,  and 
January  2d,  1854  ? 


120.  What  is  the  difference  between  two  denominate  numbers? 
Give  the  rule  for  subtraction.  How  do  you  prove  subtraction  ? 

131.  Give  the  rule  for  finding  the  difference  between  two  date*-  How 
is  the  month  reckoned  ?  At  what  time  docs  a  civil  day  begin  ? 


126  SUBTRACTION   OF 

5.  What  time  elapsed  between  November  8th,  at  1 1  o'clock 
A.M.,  1847,  and  December  16th,  at  4  o'clock,  P.M.,  1850  ? 

OPERATION. 

ANALYSIS. — The  hours  are  numbered       1/r     vnn     fjn      i>r 

_  y  /  •         //tC/.       tit/.         /If  . 

from  12  at  night,  when  the  civil  day  begins.     1359      10      IA      i/» 
The  numbers  of  the  years,  months,  days     184* 
and  hours  are  used. 

3185 

6.  What  time  elapsed  between  October  9th,  at  11  P.M., 
1840,  and  February  6th,  at  9  P.M.,  1853  ? 

7.  Mr.  Johnson  was  born  September  6th,  1771,  at  9  o'clock 
A.M.,  and  his  first  child  November  5th,  1801,  at  9  o'clock 
P.M. :  what  was  the  difference  of  their  ages  ? 

APPLICATIONS   IN    ADDITION    AND    SUBTRACTION. 

1.  From  38mo.  2wk.  Zda.  7/ir.  10m.,  take  lOmo.   Zwk. 
2da.  Whr.  50m. 

2.  From  176t/r.  8mo.  3wh  4da.,  take  91yr.  9mo. 


3.  From  £3,  take  3s. 

4.  From  2/6.  take  20#r.  Troy. 

5.  From  8R,  take  lft>  1  3  23  23. 

6.  From  9T.r  take  IT.  lewt.  2qr.  20/6.  15o2. 

7.  From  3  miles,  take  3/wr.  19rd. 

8.  The  revolution  commenced  April  19th,   1775,  and  a 
general  peace  took  place  January  20,  1783  :  how  long  did 
the  war  continue  ? 

9.  America  was   discovered  by   Columbus,    October   11, 
1492 :  what  was  the  length  of  time  to  July  25,  1855  ? 

10.  I  purchased   167/6.  8oz.   IGpwt.  lOgrr.  of  silver,  and 
sold  98/6.  lOoz.  I2frwt.  Wgr.  :  how  much  had  I  left? 

11.  I  bought  19T.   llcwt.   Zqr.   2/6.   12oz.,  12c?r.  of  old 
,'ron,  and  sold  17  T.  IScwt.  2^r.  19/6.  14oz.  lOc^r. :  what  had 

I  left  ? 

12.  I  purchased  lOlIbll?   ^3  23   19pr.  of  medicine, 
and  sold  17ft>2333    1&  bgr.:  how  much  remained  un- 
sold? 

13.  From  46?/d.  Iqr.  3na.,  take  42^.  3qr.  Ina.  2m. 

14.  Bought  7  cords  of  wood,  and  2  cords  78  feet  having 
been  stolen,  how  much  remained  ? 


DENOMINATE   NUMBERS.  157 

.5.  A  owes  B  £100 :  what  will  remain  due  after  he  has 
paid  him  £25  3s.  6J<*.  ? 

16.  A  farmer  raised   136  bushels  of  wheat ;   if  he  sells 
496w.  2p.  Iqt.  Ipt.,  how  much  will  he  have  left? 

17.  From  174/iM.  Wgal.  Iqt.  Ipt.  of  beer,  take  SQhhd. 
17 gals.  2qt.  Ipt.  * 

18.  A  farmer  had  5766w.   Ipk.  %qt.  of  wheat ;    he  sold 
1396w.  2p&.  3qt.  Ipt. :  how  much  remaiued  unsold? 

19.  A  merchant  bought   Vlcwt.  2qr.   14/6.  of  sugar,  of 
which  he  sold  at  one  time  3cwt.  Zqr.  20/6. ;  at  another  Qcwt. 
Iqr.  5/6. :  how  much  remained  unsold  ? 

20.  Sold  a  merchant  one  quarter  of  beef  for  £2  7s.  9d  ; 
one  cheese  for  9s  Id. ;  20  bushels  of  corn  for  £4  10s.  lid. ; 
and  40  bushels  of  wheat  for  £19  12s.  8Jd.  :  how  much  did 
the  whole  come  to  ? 

21.  Bought  of  a  silversmith  a  teapot,  weighing  3/6.  4oz. 
Qpivt.  2lgr.  ;  one  dozen  of  silver  spoons,  weighing  2/6.  loz. 
Ipwt.  ;   2  dishes  weighing  16/6.  lOoz.   ISpwt.  IQgr.  :    how 
much  did  the  whole  weigh  ? 

22.  Bought  one  hogshead  of  sugar  weighing  $cwt.  3qr.  2/6. 
14oz. ;    one   barrel  weighing  3cwt.  Iqr.  2/6.,  and  a  second 
barrel  weighing  Scwt.  Qqr.  lib.  4oz. :    how  much  did  the 
whole  weigh? 

23.  A  merchant  buys  two  hogsheads  of  sugar,  one  weigh- 
ing Scwt.  3qr.  21/6.,  the  other  9cwt.  2qr.  6/6.  ;  he  sells  two 
barrels,  one  weighing  3cwt.  Iqr.  12/6.  14oz.,  the  other,  Zcwt. 
Bqr.  15/6.  6oz. :  how  much  remains  on  hand  ? 

24.  A  man  sets  out  upon  a  journey  and  has  200  miles  to 
travel ;  the  first  day  he  traveled  9  leagues  2  miles  7  furlongs 
30  rods  ;  the  second  day  12  leagues  1  mile  1  furlong  ;  the 
third  day  14  leagues  ;    the  fourth  day  15  leagues  2  miles  ^ 
5  furlongs  35  rods :  how  far  had  he  then  to  travel  ? 

25.  A  farmer  has  two  meadows,  one  containing  §A.  ZR. 
37P.,  the  other  contains  10A   2R.  25P.  ;  also  three  pas-  , 
tures,  the  first  containing  12^4.  IE.  IP.  ;  the  second  con-' 
taining  13A  BE.,  and  the  third  &A.  IE.  39P.  :    by  how 
many  acres  does  the  pasture  exceed  the  meadow  land  ? 

26.  Supposing  the  Declaration  of  Independence  to  have 
been  published  at  precisely  12  o'clock  on  the  4th  of  July, 
1776,  how  much  time  elapsed  to  the  1st  of  January,  1833, 
at  25  minutes  past  3,  T.M.  ? 


128  MULTIPLICATION   OF 


MULTIPLICATION  OF  DENOMINATE  NUMBERS. 

122.  MULTIPLICATION  of  denominate  numbers  is  the  opera- 
tion of  multiplying  a  denominate  number  by  an  abstract  number. 

I.  A  tailor  has  5  pieces  of  cloth  each  containing  6yd~ 
%qr.  3na. :  how  many  yards  are  there  in  all  ? 

ANALYSIS. — In    all    the    pieces    there   are   5    OPERATION. 
times  as  much  as  there   is  in  1  piece.     If  in     yd.     or.     na. 
1  piece   each   denomination    be    taken  5  times,       it       o        3 
the  result  will  be  5  times  as  great  as  the  multi- 
plicand.   Taking    each     denomination   5   times, 

we  have  30#d.  lO^r.  15?ia.  30     10     15" 

But,  instead  of  writing  the  separate  products,     33        1        3 
we    begin   with   the   lowest    denomination   and 
say,  5  times  3na.  are  15na. ;  divide  by  4,  the  units  of  the  scale,  write 
down  the  remainder  3fta.,  and  reserve  the  quotient  Sgr.  for  the 
next  product.     Then  say,  5  times  2qr.  are  10§r.,  to  which  add  the 
%qr.  making  13gr.     Then  divide  by  4,  write  down  the  remainder 
1,  and  reserve  the  quotient  3  for  the  next  product.     Then  say,  5 
times  6  are  30,  and  3  to  carry  are  33  yards :  hence, 

RULE. — I.   Write  down  the  denominate  number  and  set 
the  multiplier  under  the  lowest  denomination. 

II.  Multiply  as  in  simple  numbers,  and  in  passing  from  one 
denomination  to  another,  divide  by  the  units  of  the  scale,  set 
down  the  remainder  and  carry  the  quotient  to  the  next  product. 

PROOF. — The  same  as  in  simple  numbers. 


£ 
17 

CM. 

s.   d 

15  9 

.far. 
6 

EXAMPLES. 
T. 

c?r/. 
10 

(2. 
*£ 

:>* 

2 

oz. 
12 

7 

106  14  10 

(3.) 
m.fur.  rd. 
9  3  20 

2       3 

*? 

6 

10 

8. 

9 

0 

0 

9 

19 

(4.) 

27 

4 

35 
3 

132.  What  is  multiplication  of  denominate  numbers?    Give  the  rule. 
How  do  you  prove  multiplication  ? 


DENOMINATE   NUMBERS.  129 

(5.)  (6.) 

yr.  mo.  da.    hr.  T.   cwt.  qr.    Ib.  oz.  dr. 

6     5     15     18  6     12     3     20  12     9 
5                                                       8 


7.  A  farmer  has  11  bags  of  corn,  each  containing  26w.  Ipk. 
3qt.  :  how  much  corn  in  all  the  bags  ? 

8.  How  much  sugar  in  12  barrels,  each  containing  3cw£ 
3qr.  2/6.  ? 

9.  In  7  loads  of  wood,  each  containing  1  cord  and  2  cord 
feet,  how  many  cords  ? 

10.  A  bond  was  given  21st  of  May,  1825,  and  was  taken 
up  the  12th  of  March,  1831 ;  what  will  be  the  product,  if 
the  time  which  elapsed  from  the  date  of  the  bond  till  the  day 
it  was  taken  up,  be  multiplied  by  3  ? 

11.  What  is  the  weight  of'l  dozen  silver  spoons,  each 
weighing  3oz.  Spwt.  ? 

12.  What  is  the  weight  of  7  tierces  of  rice,  each  weighing 
5cwt.  2qr.  16/6.?   ' 

13.  Bought  4  packages  of  medicine,  each  containing  3fi> 
4^   63   13  16#r. :  what  is  the  weight  of  all  ? 

14.  How  far  will  a  man  travel  in  5  days  at  the  rate  of 
24mi.  4/ur.  krd.  per  day  ? 

15.  How  much  land  is  there  in  9  fields,  each  field  contain- 
ing 12^.  IK  25P.? 

16.  How  many  yards  in  9  pieces,  each  29 yd.  2qr.  3na.  ? 

17.  If  a  vessel   sails   5L.  2>mi.  6fur.  SQrd.  in  one  day, 
how  far  will  it  sail  in  8  days  ? 

18.  How  much  water  will  be  contained  in  96  hogsheads, 
each  containing  QZgal.  Iqt.  Ipt.  Igi.  ? 

NOTE. — When  the  multiplier  is  a  composite  number,  and  the 
factors  do  not  exceed  12,  multiply  by  the  factors  in  succession. 
In  the  last  example  96=12  x  8. 

19.  If  one  spoon  weighs  3oz.   5pwt.    15<?r.  what  is   the 
weight  of  120  spoons?  i 

20.  If  a  man  travel  249m.  7/itr.  4rd.  in  one  day,  how  far 
will  he  go  in  one  month  of  30  days? 

21.  If  the  earth  revolve  0°  15'  of  space  per  minute  of  tune, 
how  far  does  it  revolve  per  hour  ? 

22.  Bought  90/i/id.  of  sugar,  each  weighing  IZcwt.  Zqr. 
. :  what  was  the  weight  of  the  whole? 


130  DIVISION   OF 

23.  What  is  the  cost  of  18  sheep,  at  5s.  9|d.  apiece  ? 

24.  How  much  molasses  is  contained  in  2bhhd.  each  hogs- 
head having  ftlgal.  \qt.  Ipt.  ? 

25.  How  many  yards  of  cloth  in  36  pieces,  each  piece  con- 
taining %5yd.  3qr.  ? 

26.  A  farmer  has  18  lots,  and  each  lot  contains  41 A  2#. 
IIP. :  how  many  acres  does  he  own? 

21.  There  are  three  men  whose  mutual  ages  are  14  times 
2Qyr.  5mo.  3wk.  Qda. :  what  is  the  sum  of  their  ages? 

28.  Bought  90/i/id.  of  sugar,  each  weighing   12cwt.   2qr. 
14lb. ;  what  is  the  weight  of  the  whole  ? 

29.  If  a  vessel  sail  49ml  Qfur.  8rd.  in  one  day,  how  far 
will  she  sail  in  one  month  of  30  days  ? 

30.  Suppose  each  of  50  farmers  to  raise  125£m.  3pk.  6qt.  of 
grain  :  how  much  do  they  all  raise  ? 

31.  If  a  steam  ship,  in  crossing  the  Atlantic,  goes  211mi. 
4/wr.  32rd.  a  day,  how  far  will  she  go  in  15  days? 

32.  If  1  horse  consume  2  tons  Iqr.  20/6.  of  hay  in  a  winter, 
how  much  will  36  horses  consume? 

33.  How  much  cloth  will  clothe  a  company  of  48  men,  if 
it  takes  5yd.  3qr.  2na.  to  clothe  one  man  ? 

NOTE. — Each  denomination  may  Be  multiplied  by  the  multiplier, 
separately,  and  the  results  reduced  and  added. 


DIVISION  OF  DENOMINATE  NUMBERS. 

123.  DIVISION  of  denominate  numbers  is  the  operation  of 
dividing  a  denominate  number  into  as  many  equal  parts  as 
there  are  units  in  the  divisor. 

1.  Divide  £25  15s.  4d.  by  8. 

ANALYSIS. — We  first  say  8  into  25,  3  times  OPERATION. 

and  £1  or  20s.  over.     Then,  after  adding  the  8)d£25   15s.  la 

15s.  we  say,  8  into  35,  4  times  and  3s.  over.  — ^ — -j F~? 

Then,  reducing  the  3*.  to  pence  and  adding  in 
the  4rf.,  we  say,  8  into  40,  5  times. 

123.  What  is  division  of  denominate  numbers?  Give  the  rule  for 
division.  How  do  you  prove  division  ?  How  do  you  divide  when  the 
divisor  is  a  composite  number  ?  What  will  be  the  unit  of  each  quo- 
tient figure  ? 


DENOMINATE    NUMBERS.  131 

OPERATION. 

t366w. 
2.  Divide  366tt.  3pfc.  Iqt.  by  7. 

ANALYSIS.— In  this  example  we 
find  that  7  is  contained  in  36  bushels 
5  times  and  1  bushel  over.  Reducing 
this  to  pecks,  and  adding  3  pecks, 
gives  7  pecks,  which  contains  7,  1 
time  and  no  remainder.  Multiplying 
0  by  8  quarts  and  adding,  gives  7 
quarts  to  be  divided  by  7. 

7)7(lqt 


Ans.  bbu.  \pk.  Iqt. 

Hence,  for  the  division  of  denominate  numbers  we  have  the 
following 

RULE. — I.  Begin   with    the    highest,    denomination  and 
divide  as  in  simple  numbers  : 

II.  Reduce  the  remainder,  if  any,  to  the  next  lower  de- 
nomination, and  add  in  the  units  of  that  denomination  for 
a  new  dividend. 

III.  Proceed  in  the  same  manner  through  all  the  denomi- 
nations. 

PROOF. — By  multiplication,  as  in  simple  numbers. 

NOTES. — 1.  If  the  divisor  is  a  composite  number,  we  may  divide 
by  the  factors  in  succession,  as  in  simple  numbers. 

2.  Each  quotient  figure  has  the  same  unit  as  the  dividend  from 
which  it  was  derived. 

3.  If  the  divisor  is  greater  than  12  and  not  a  composite  number, 
the  operation  is  the  same  as  long  division. 

EXAMPLES. 

(1.)  (2.) 

T.    cwt.  qr.   Ib.  A.      It.  P. 

7)1     19     2     12  9)113     3  25 

Quotient.   "              5216  122  25 

(3.)  (4.) 

L.     mi.  fur.  rd.  bu.      pk.  qL 

8)47     1      7  8  11)25     3     1 

Quotient. 


132  DIVISION  OF 


Divide  the  following : 

5.  l*lcwt.  Qqr.  2/6.  6oz.  by  7. 

6.  49*/d.  3?r.  3/m.  by  9. 

7.  131A  1,8.  by  12. 


8.  £1138  12s.  4a.  by  53. 


9.  TOT.  17cwtf.  7/6.  by  79. 
10.  276u.  Spyfc.  7^.  by  84. 


11.  Bought  65  yards  of  cloth  for  which  I  paid  £72  14s. 
. :  what  did  it  cost  per  yard? 

12.  If  15  loads  of  hay  contain  35  T.  5cwt.,  what  is  the 
weight  of  each  load  ? 

13.  If  a  man,  lifting  8  times  as  much  as  a  boy,  can  raise 
201/6.  12oz.,  how  much  can  the  boy  lift? 

14.  If  a  vessel  sail  25°  42'  40"  in  10  days,  how  far  will 
she  sail  in  one  day  ? 

15.  Divide  Vhhd.  ZZgal.  2qt.  by  12. 

16.  What  is  the  quotient  of  656w.  Ipk.  3qt.  divided  by  12? 

17.  In  4  equal  packages  of  medicine  there  are  13B>   7  3 
23   13  4gr. ;  how  much  is  there  in  each  package  ? 

18.  In  25hhd.  of  molasses,  the  leakage  has  reduced  the 
whole  amount  to  1534gra/.  \qt.  \pt. :  if  the  same   quantity 
has  leaked  out  of  each  hogshead,  how  much  will  each  hogs- 
head still  contain  ? 

19.  In  9  fields  there  are  113A  37?.  25P.  of  land  :  if  the 
fields  contain  an  equal  amount,  how  much  is  there  in  each 
field? 

20.  If  in  30  days  a  man  travels  746mi.  5/wr.,  travelling 
the  same  distance  each  day,  what  is  the  length  of  each  day's 
journey  ? 

21.  Suppose  a  man  had  98/6.  2oz.  Wpwt.  6gr.  of  silver ; 
how  much  must  he  give  to  each  of  7  men  if  he  divides  it 
equally  among  them? 

22.  When   J75#a/.  2qt.  of  beer  are  drank  in  52  weeks, 
how  much  is  consumed  in  one  week  ? 

23    A  rich  man  divided  1686w.  Ipk.  Qqt.  of  corn  among 
35  poor  men  :  how  much  did  each  receive  ? 

24.  In  sixty-three  barrels  of. sugar  there  are  7T.  16cwtf. 
3qr.  12/6. :  how  much  is  there  in  each  barrel  ? 

25.  A  farmer   has  a  granary  containing   232  bushels  3 
ks  7  quarts  of  wheat,  and  he  wishes  to  put  it  in  105  bags  : 

ow  much  must  each  bag  contain  ? 

26.  If  90  hogsheads  of  sugar  weigh  56 T.  Hcwt.  Zqr.  15/6, 
what  u  the  weight  of  1  hogshead  ? 


DENOMINATE  NUMBERS.  133 

27.  One  hundred  and  seventy-six  men  consumed  in  a  week 
IScwt    2qr.   15/6.  6oz.  of  bread :   how  much  did  each  man 
consume  ? 

28.  If  the  earth  revolves  on  its  axis  15°  in  1  hour,  how  far 
does  it  revolve  in  1  minute  ? 

29.  If  59  casks  contain  44Md.  ttgal.  2qt.  Ipt.  of  wine, 
what  are  the  contents  of  one  cask  ? 

30.  Suppose  a  man  has  246ml  Qfur.  36rd.  to  travel  in  12 
days  :  how  far  must  he  travel  each  day? 

31.  If  I  pay  £12  14s.  5d  3/ar.  for  35  bushels  of  wheat, 
what  is  the  price  per  bushel  ? 

32.  A  printer  uses  one  sheet  of  paper  for  every  16  pages  of 
an  octavo  book :  how  much  paper  will  be  necessary  to  print 
500  copies  of  a  book  containing  336  pages,  allowing  2  quires 
of  waste  paper  in  each  ream  ?* 

33.  A  man  lends  his  neighbor  £135  6s.  8d.,  and  takes  in 
part  payment  4  cows  at  £5  8s.  apiece,  also  a  horse  worth 
£50  :  how  much  remained  due  ? 

34.  Out  of  a  pipe  of  wine,  a  merchant  draws  12  bottles, 
each  containing  1  pint  3  gills  ;  he  then  fills  six  5-gallon  demi- 
johns ;  then  he  draws  off  3  dozen  bottles,  each  containing 
1  quart  2  gills :  how  much  remained  in  the  cask  ? 

35.  A  farmer  has  6  T.  Scivt.  2qr.  14/6.  of  hay  to  be  re- 
moved in  6  equal  loads :  how  much  must  be  carried  at  each 
load? 

36.  A  person  at  his  death  left  landed  estate  to  the  amount 
of  £2000,  and  personal  property  to  the  amount  of  £2803  17s. 
4c?.     He  directed  that  his  widow  should  receive  one-eighth  of 
the  whole,  and  that  the  residue  should  be  equally  divided 
among  his  four  children :   what  was  the  widow  and  each 
child's  portion  ? 

37.  If  a  steamboat  go  224  miles  in  a  day,  how  long  will 
it  take  to  go  to  China,  the  distance  being  about  12000  miles? 

38.  How  long  would  it  take  a  balloon  to  go  from  the  earth 
to  the  moon,  allowing  the  distance  to  be  about  240000  miles, 
the  balloon  ascending  34  miles  per  hour  ? 


*  In  packing  and  selling  paper,  the  two  outside  quires  of  every  ream 
are  regarded  as  waste,  and  each  of  the  remaining  quires  contains  34 
perfect  sheets:  hence,  in  this  example,  the  waste  "paper  is  considered 
as  belonging  only  to  the  entire  reams. 


134  LONGITUDE   AND   TIME. 


LONGITUDE  AND  TIME. 

124.  The  circumference  of  the  earth,  like  that  of  other 
circles,  is  divided  into  360°,  which  are  called  degrees  of  lon- 
gitude. 

125.  The  sun  apparently  goes  round  the  earth  once  in  24 
hours.     This  time  is  called  a  day. 

Hence,  in  24  hours,  the  sun  apparently  passes  over  360°  of 
longitude  ;  and  in  1  hour  over  360°  -=-24  =  15°. 

126.  Since  the  sun,  in  passing  over  15°  of  longitude,  re- 
quires  1  hour  or  GO'  of  time,  1°  will  require  60'-=- 15  =  4= 
minutes  of  time  ;  and  V  of  longitude  will  be  equal  to  one 
sixteenth  of  4'  which  is  4"  :  hence, 

15°  of  longitude  require  1  hour 
1°  of  longitude  requires  4  minutes. 
1'  of  longitude  requires  4  seconds. 

Hence,  we  see  that, 

1.  If  the  degrees  of  longitude  be  multiplied  by  4,  the  pro- 
duct will  be  the  corresponding  time  in  minutes. 

2.  If  the  minutes  in  longitude  be  multiplied  by  4,  the  pro- 
duct will  be  the  corresponding  time  in  seconds. 

127.  When  the  sun  is  on  the  meridian  of  any  place,  it  is 
12  o'clock,  or  noon,  at  that  place. 

Now,  as  the  sun  apparently  goes  from  east  to  west,  at  the 
instant  of  noon,  it  will  be  past  noon  for  all  places  at  the  east, 
and  before  noon  for  all  places  at  the  west. 

If  then,  we  find  the  difference  of  time  between  two  places, 
and  know  the  exact  time  at  one  of  them,  the  corresponding 
time  at  the  other  will  be  found  by  adding  their  difference,  if 
that  the  other  be  east,  or  by  subtracting  it  if  west. 


124.  How  is  the  circumference  of  the  earth  supposed  to  be  divided  ? 

125.  How  does  the  sun  appear  to  move  ?    What  is  a  day  ?    How  far 
does  the  sun  appear  to  move  in  1  hour  ? 

126.  How  do  you  reduce  degrees  of  longitude  to  time  ?    How  do  you 
reduce  minutes  of  longitude  to  time  ? 

127.  What  is  the  hour  when  the  sun  is  on  the  meridian  ?    When  the 
sun  is  on  the  meridian  of  any  place,  how  will  the  time  be  for  all  places 
cast?    How  for  all  places  west?     If  you  have  the  difference  of  time, 
how  do  you  find  the  time  V 


LONGITUDE   AND   TIME.  135 

1.  The  longitude  of  New  York  is  74°  1'  west,  and  that  of 
Philadelphia  75°  10'  west :  what  is  the  difference  of  longi- 
tude and  what  their  difference  of  time  ? 

2.  At  12  M.  at  Philadelphia,  what  is  the  time   at  New 
York? 

3.  At  12  M.  at  New  York,  what  is  the  time  at  Philadelphia  ? 

4.  The  longitude  of  Cincinnati,  Ohio,   is  84°   24'  west : 
what  is  the  difference  of  time  between  New  York  and  Cin- 
cinnati ? 

5.  What  is  the  time  at  Cincinnati,  when  it  is  12  o'clock  at 
New  York? 

6.  The  longitude  of  New  Orleans  is  89°  2'  west :  what 
time  is  at  New  Orleans,  when  it  is  12  M.  at  New  York  ? 

7.  The  meridian  from  which  the  longitudes  are  reckoned 
passes  through  the  Greenwich  Observatory,  London  :  hence, 
the  longitude  of  that  place  is  0  :  what  is  the  difference  of 
time  between  Greenwich  and  New  York  ? 

8.  What  is  the  time  at  Greenwich,  when  it  is  12  M.  at 
New  York? 

9.  The  longitude  of  St.  Louis  is  90°  15'  west :  what  is  the 
time  at  St.  Louis,  when  it  is  3/i.  25m.  P.M.  at  New  York  ? 

10.  The  longitude  of  Boston  is  71°  4'  west,  and  that  of 
New  Orleans  89°  2'  west :  what  is  the  time  at  New  Orleans 
when  it  is  7  o'clock  12??i  A.M.  at  Boston  ? 

11.  The  longitude  of  Chicago,  Illinois,   is   87°  30'  west : 
what  is  the  time  at  Chicago,  when  it  is  12  M.  at  New  York? 

PROPERTIES  OF  NUMBERS. 

COMPOSITE    AND    PRIME    NUMBERS. 

128.  An  Integer,  or  whole  number,  is  a  unit  or  a  collection 
of  units. 

129.  One  number  is  said  to  be  divisible  by  another,  when 
the  quotient  arising  from  the  division  is  a  whole  number.  The 
division  is  then  said  to  be  exact. 

NOTE. — Since  every*  number  is  divisible  by  itself  and  1,  the 
term  divisible  will  be  applied  to  such  numbers  only,  as  have  other 
divisors. 

128.  What  is  an  Integer  ? 


136  PROPERTIES   OF  NUMBERS. 

130.  Every  divisible  number  is  called  a  composite  number, 
(Art.  54),  and  any  divisor  is  called  &  factor:  thus,  6  is  a  com- 
posite number,  and  the  factors  are  2  and  3. 

131.  Every  number  which  is  not  divisible  is  called  a  prime 
number :  thus,  1,  2,  3,  5,  7,  11,  &c.  are  prime  numbers. 

132.  Every  prime  number  is  divisible  by  itself  and  1  ; 
but  since  these  divisors  are  common  to  all  numbers,  they  are 
not  called  factors. 

133.  Every  factor  of  a  number  is  either  prime  or  compo- 
site :  and  since  any  composite  factor  may  be  again  divided,  it 
follows  that, 

Any  number  is  equal  to  the  product  of  all  its  prime  factors. 

For  example,  12=: 6  x  2  ;  but  6  is  a  composite  number,  of 
which  the  factors  are  2  and  3  ;  hence, 

12=2  x  3  x  2  ;  also,  20=10  x  2=5  x  2  x  2. 
Hence,  to  find  the  prime  factors  of  any  number, 

Divide  the  number  by  any  prime  number  that  will  exactly 
divide  it :  then  divide  the  quotient  by  any  prime  number  that 
will  exactly  divide  it,  and  so  on,  till  a  quotient  is  found  which 
is  a  prime  number  ;  the  several  divisors  and  the  last  quotient 
will  be  the  prime  factors  of  the  given  number. ' 

NOTE. — It  is  most  convenient,  in  practice,  to  use  the  least  prime 
number,  which  is  a  divisor. 

1.  What  are  the  prime  factors  of  42  ? 

OPERATION. 

ANALYSIS. — Two    being   the    least    divisor  2)42 

that  is  a  prime  number,  we  divide  by  it,  giv-  o\91 

ing  the   quotient  21,  which  we  again  divide  o)4L 

by  3,  giving  7:    hence,  2,  3  and  7   are  the  7 

prime  factors.  2x3x7  =  42. 


129.  When  is  one  number  divisible  by  another  ?    By  what  is  every 
number  divisible  ?    Is  1  called  a  divisor  ? 

130.  What  is  a  composite  number  ?    What  is  a  factor  ? 

131.  What  is  a  prime  number  ? 

132.  By  what  divisors  is  every  prime  number  divided  ? 

133.  To  what  product  is  every  number  equal?    Give  the  rule  for 
finding  the  prime  factors  of  a  number.     What  number  is  it  most  conve- 
nient to  use  as  a  divisor  ? 


PRIME   FACTORS.  137 

What  arc  the  prime  factors  of  the  following  numbers  ? 


1.  Of  the  number  9  ? 

2.  Of  the  number  15? 

3.  Of  the  number  24  ? 

4.  Of  the  number  16? 

5.  Of  the  number  18  ? 


6.  Of  the  number  32  ? 

7.  Of  the  number  48  ? 

8.  Of  the  number  56? 

9.  Of  the  number  63  ? 
10.  Of  the  number  76? 


NOTE. — The  prime  factors,  when  the  number  is  small,  may 
generally  be  seen  by  inspection.  The  teacher  can  easily  multiply 
the  examples. 

134.  When  there  are  several  numbers  whose  prime  factors 
are  to  be  found, 

Find  the  prime  factors  of  each  and  then  select  those  factors 
which  are  common  to  all  the  numbers. 

11.  What  are  the  prime  factors  common  to  6,  9  and  24  ? 

12.  What  are  the  prime  factors  common  to  21,  63  and  84? 

13.  What  are  the  prime  factors  common  to  21,  63  and  105  ? 

14.  What  are  the  common  factors  of  28,  42  and  70  ? 

15.  What  are  the  prime  factors  of  84,  126  and  210  1 

16.  What  are  the  prime  factors  of  210,  315  and  525  ? 

135.    DIVISIBILITY    OF    NUMBERS. 

1.  2  is  the  only  even  number  which  is  prime. 

2.  2  divides  every  even  number  and  no  odd  number. 

3.  3  divides  any  number  when  the  sum  of  its  figures  is  di- 
visible by  3. 

4.  4  divides  any  number  when  the  number  expressed  by 
the  two  right  hand  figures  is  divisible  by  4. 

5.  5  divides  every  number  which  ends  in  0  or  5. 

6.  6  divides  any  even  number  which  is  divisible  by  3. 

7.  10  divides  any  number  ending  in  0. 

GREATEST  COMMON  DIVISOR. 

130.  The  greatest  common  divisor  of  two  or  more  num- 
bers, is  the  greatest  number  which  will  divide  each  of  them, 
separately,  without  a  remainder.  Thus,  6  is  the  greatest 
common  divisor  of  12  and  18. 


134.  How  do  you  find  the  prime  factors  of  two  or  more  numbers  ? 


138  COMMON   DIVISOR. 

NOTE. — Since  1  divides  every  number,  it  is  not  reckoned  among 
the  common  divisors. 

137.  If  two  numbers  have  no  common  divisor,  they  are 
called  prime  with  respect  to  each  other. 

138.  Since  a  factor  of  a  number  always  divides  it,  it  fol- 
lows that  the  greatest  common  divisor  of  two  or  more  num- 
bers, is  simply  the  greatest  factor  common  to  these  numbers. 

Hence,  to  find  the  greatest  common  divisor  of  two  or 
more  numbers, 

I.  Resolve  each  number  into  its  prime  factors. 

II.  The  product  of  the  factors  common  to  each  result  will 
be  the  greatest  common  divisor. 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  24  and  30  ? 

ANALYSIS. — There  are  four  prime  OPERATION. 

factors  in  24,  and  3  in  30  :  the  factors          24  =  2x2x2x3 
2  and  3  are  common  :  hence,  6  is  the          30  =  2  X  3  X  5 

greatest  common  divisor.  2X^(>  com.  divisor. 

2.  What  is  the  greatest  common  divisor  of  9  and  18  ? 

,  3.  What  is  the  greatest  common  divisor  of  6,  12,  and  30  ? 

4.  What  is  the  greatest  common  divisor  of  15,  25  and  30  ? 

5.  What  is  the  greatest  common  divisor  of  12,  18  and  72  ? 

6.  What  is  the  greatest  common  divisor  of  25,  35  and  70  ? 

7.  What  is  the  greatest  common  divisor  of  28,  42  and  70  ? 

8.  What  is  the  greatest  common  divisor  of  84,  126  and 
210? 

139.  When  the  numbers  are  large,  another  method  of  find- 
ing their  greatest  common  divisor  is  used,  which  depends  ou 
the  following  principles : 


135.  What  even  number  is  prime  ?    What  numbers  will  2  divide  ? 
What  numbers  will  3  divide  ?    What  numbers  will  4  divide  ?    5  ?    6  ? 
10? 

136.  What  is  the  greatest  common  divisor  of  two  or  more  numbers  ? 

137.  When  are  two  numbers  said  to  be  prime  with  respect  to  each 
other? 

138.  What  is  the  greatest  factor  of  two  numbers  ?    How  do  you  find 
the  greatest  common  divisor  of  two  or  more  numbers  ? 


PROPERTIES   OF   NUMBERS.  139 

1.  Any  number  which  willdividetwo  numbers  separately,  will 
divide  their  sum  ;  else,  we  should  have  a 

whole  number  equal  to  a  proper  fraction.       24+27=51 

2.  Any  number  which  will  divide  two  numbers  separately, 
ivill   divide   their  difference;    and   any 

number  which  will  divide  their  differ-       51  —  27  =  24 
ence  and  one  of  the  numbers,  will  divide 
the  other  ;  else,  we  should  have  a  whole  number  equal  to  a 
proper  fraction. 


1. 


*/ 

What  is  the  greatest  common  divisor  of  27  and  51  ? 


Divide  51  by  27 ;  the  quotient  is  1  and  the  remainder  24 ;  then 

divide  the  preceding  divisor  27  by  the  re-  OPERATION. 

mainder  24  :  the  quotient  is  1  and  the  re  27)51(1 

mainder    3 :    then    divide   the  preceding  27 

divisor  24  by  the  remainder  3  ;  the  quo-  

tient  is  8  and  the  remainder  0.  24 ) 27  ( 1 

Now,  since  3  divides  the  difference  3, 


and  also  24,  it  will  divide  27,  by  principle  3)24(8 

2d  ;  and  since  3  divides  the  remainder  24,  04 

and  27,  it  will  also  divide  51 :  hence  it  is 

a  common  divisor  of  27  and  51  ;  and  since  it  is  the  greatest  com- 
mon factor,  it  is  their  greatest  common  divisor.  Since  the  above 
reasoning  is  as  applicable  to  any  other  two  numbers  as  to  27  and 
51,  we  have  the  following  rule  : 

Divide  the  greater  number  by  the  less,  and  then  divide  the 
preceding  divisor  by  the  remainder,  and  so  on,  till  nothing  re- 
mains :  the  last  divisor  will  be  the  greatest  common  divisor. 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  216  and  408  ? 

2.  Find  the  greatest  common  divisor  of  408  and  740. 

3.  Find  the  greatest  common  divisor  of  315  and  810. 

4.  Find  the  greatest  common  divisor  of  4410  and  5670. 

5.  Find  the  greatest  common  divisor  of  3471  and  1869. 

6.  Find  the  greatest  common  divisor  of  1584  and  2772. 

NOTE. — If  it  be  required  to  find  the  greatest  common  divisor  of 
more  than  two  numbers,  first  find  the  greatest  common  divisor  of 

139.  When  the  numbers  are  large,  on  what  principles  docs  the  oper- 
ation of  finding  the  greatest  common  divisor  depend  ?  What  is  the 
rule  for  finding  it  ? 


140  COMMON   MULTIPLE* 

two  of  them,  then  of  that  common  divisor  and  one  of  the  remain 
ing  numbers,  and  so  on  for  all  the  numbers  ;  the  last  common 
divisor  will  be  the  greatest  common  divisor  of  all  the  numbers. 

7.  What  is  the  greatest  common  divisor  of  492,  744  and 
1044? 

8.  What  is  the  greatest  common  divisor  of  944,  1488,  and 
2088? 

9.  What  is  the  greatest  common  divisor  of  216,  408  and 
740? 

10.  What  is  the  greatest  common  divisor  of  945  1560  and 
22683  ? 

LEAST  COMMON  MULTIPLE. 

140.  The  common  multiple,  of  two  or  more  numbers,  is  any 
number  which  will  exactly  divide. 

The  least  common  multiple  of  two  or  more  numbers,  is  the 
least  number  which  they  will  separately  divide  without  a  re- 
mainder. 

NOTES. — 1.  If  a  dividend  is  exactly  divisible  by  a  divisor,  it  can 
be  resolved  into  two  factors,  one  of  which  is  the  divisor  and  the 
other  the  quotient. 

2.  If  the   divisor   be   resolved   into  its  prime   factors,  the  cor- 
responding factor  of  the  dividend  may  be  resolved  into  the  same 
factors :  hence,  the  dividend  will  contain  every  prime  factor  of  the 
divisor. 

3.  The  question  of  finding  the  least  common  multiple  of  several 
numbers,  is  therefore  reduced  to  finding  a  number  which  shall  con- 
tain all  their  prime  factors  and  none  others. 

1.  Let  it  be  required  to  find  the  least  common  multiple  of 
6,  8  and  12. 

ANALYSIS. — We  see,  from  inspec-  OPERATION. 

tion,  that  the  prime  factors  of  6  are  2x3      2x2x2      2x2x3 

2  and  3 :— of  8 ;   2,  2  and  2 :— and         6 8 12 

of  12  ;  2,  2  and  3. 

Every  number  that  is  a  prime  factor  must  appear  in  the  least  com- 
mon multiple,  and  none  others:  hence,  it  will  contain  all  the  prime 

140.  What  is  the  least  common  multiple  of  two  or  more  numbers  ? 
State  the  principles  involved  in  finding  it.  Give  the  rule  for  finding  it. 
What  is  the  multiple  when  the  numbers  have  no  common  prime  fac- 
tors ? 


COMMON   MULTIPLE.  141 

factors  of  any  one  of  the  numbers,  as  8,  and  such  other  prime  fac- 
tors of  the  others,  6  and  12,  as  are  not  found  among  the  prime  fac- 
tors of  8 ;  that  is,  the  factor  3 :  hence, 

2  x  2  x  2  x  3  =  24,  the  least  common  multiple. 
To  find  the  least  common  multiple  of  several  numbers. 

I.  Place  the  numbers  on  the  same  line,  and  divide  by  any- 
prime  number  that  will  exactly  divide  two  or  more  of  them, 
and  set  down  in  a  line  below  the  quotients  and  the  undivided 
numbers. 

II.  Then  divide  as  before  until  there  is  no  prime  number 
greater  than  1  that  will  exactly  divide  any  two  of  the  numbers. 

III.  Then  multiply  together  the  divisors  and  the  numbers  of 
the  lower  line,  and  their  product  will  be  the  least  common 
multiple. 

NOTE. — 1.  The  object  of  dividing  by  any  prime  number  that  will 
divide  two  or  more  of  the  numbers,  is  to  find  common  factors.       x 

2.  If  the  numbers  have  no  common  prime  factor,  their  product 
will  be  their  least  common  multiple. 

EXAMPLES. 

OPERATION. 

1.  Find  the  least  common  mul- 
tiple of  3,  4  and  8.  2)3 4 8 


Ans.  2x2x3x1x2  =  24.  2)3 


2.  Find  the  least  common  mul- 
tiple of  3,  8  and  9.  3)3 8 9 

Ans.  3x1x8x3=72.  1 8 3 

3.  Find  the  least  common  multiple  of  6,  7,  8  and  10. 

4.  Find  tKe  least  common  multiple  of  21  and  49. 

5.  Find  the  least  common  multiple  of  2,  7,  5,  6,  and  8. 

6.  Find  the  least  common  multiple  of  4,  14,  28  and  98 

7.  Find  the  least  common  multiple  of  13  and  6. 

8.  Find  the  least  common  multiple  of  12,  4  and  7. 

9.  Find  the  least  common  multiple  of  6,  9,  4,  14  and  16. 

10.  Find  the  least  common  multiple  of  13,  12  and  4. 

11.  Find  the  least  common  multiple  of  11,  17,  19,  21,  and 


14:2  CANCELLATION. 

CANCELLATION. 

141.  CANCELLATION  is  a  method  of  shortening  Arithmeti- 
cal operations  by  omitting  or  cancelling  common  factors. 

1.  Divide  24  by  12.     First,  24  =  3  x  8  ;  and  12  =  3  x  4. 

ANALYSIS.  —  Twenty-four  divided  by  12  is  OPERATION. 

equal  to  3  x  8  divided  by  3  x  4  ;  by  cancelling  24        $  x  8 

or  striking  out  the  3's,  we  have  8  divided  by  ~~nr—  ~*  —  ~r  =  2 
4,  which  is  equal  to  2. 

142.  The  operations  in  cancellation  depend  on  two  princi- 
ples : 

1.  The  cancelling  of  a  factor,  in  any  number,  is  equivalent 
to  dividing  the  number  by  that  factor. 

2.  If  the  dividend  and  divisor  be  both  divided  by  the  same 
number,  the  quotient  will  not  be  changed. 

PRINCIPLES    AND    EXAMPLES. 

1.  Divide  63  by  21. 

ANALYSIS.—  Resolve  tlie  dividend  and  divi-         OPERATION. 
sor  into  factors,  and  then  cancel  those  which         63  _  *  x  9 


are  common. 


" 


2.  In  7  times  56,  how  many  times  8  ? 

ANALYSIS.—  Resolve  56  into  the  OPERATION. 

two  factors  7  and  8,  and  then  cancel         56x7_$x7x7 
the  8.  -g-          -J- 

3.  In  9  times  84,  how  many  times  12  ? 

4.  In  14  times  63,  how  many  times  7  ? 

5.  In  24  times  9,  how  many  times  8  ? 

6.  In  36  times  15,  how  many  times  45  ? 

ANALYSIS.—  We  see  that  9  is  a  factor  of  36 
and  45.     Divide  by  this  factor,  and  write  the        OPERATK  N. 
quotient  4  over  36,  and  the  quotient  5  below  4        3 

45.     Again,  5  is  a  factor  of  15  and  5.     Divide         $6  x  I'SJ 
15  by  5,  and  write    the    quotient  3   over  15.  —    _  =1 

Dividing  5  by  5,  reduces  the  divisor  to  1,  which  40 

need  not  be  set  down  :  hence,  the  true  quotient  $ 

4x3=12. 

141.  What  is  cancellation  ? 

143.  On  what  do  the  operations  of  rnneellitlon  depend  ? 


CANCELLATION.  143 

143.  Therefore,  to  perform  the  operations  of  cancellation  : 

1.  Resolve  the  dividend  and  divisor  into  such  factors  as 
shall  give  all  the  factors  common  to  both. 

II.  Cancel  the  common  factors  and  then  divide  the  product 
of  the  remaining  factors  of  the  dividend  by  the  product  of  the 
remaining  factors  of  the  divisor. 

NOTES. — 1.  Since  every  factor  is  cancelled  by  division,  the  quo- 
tient 1  always  takes  the  place  of  the  cancelled  factor,  but  is  omit- 
ted when  it  is  a  multiplier  of  other  factors. 

2.  If  one  of  the  numbers  contains  a  factor  equal  to  the  product 
of  two  or  more  factors  of  the  other,  they  may  all  be  cancelled. 

3.  If  the  product  of  two  or  more  factors  of  the  dividend  is  equal 
to  the  product  of  two  or  more  factors  of  the  divisor,  such  products 
may  ba  cancelled. 

4-  It  is  generally  more  convenient  to  set  the  dividend  on  the 
right  of  a  vertical  line  and  the  divisor  on  the  left. 

EXAMPLES. 

1.  What  number  is   equal  to  36  multiplied  by  13  and  the 
product  divided  by  4  times  9  ? 

ANALYSIS. — We  may  place  the  numbers  whose  OPERATION. 

product  forms  the  dividend  on  the  right  of  a  verti-  ^     #0 

cal  line,  and  those  which  form  the  divisor  on  the  A      10 
left.     We  see  that  4x9=36  ;  we  then  cancel  4,  9, 

and  36.  Ans.  13. 

2.  What     is     the     result     of    20  x  4  x  12,    divided    by 
10x16x3? 

OPERATION. 

ANALYSIS. — First,  cancel  the  factor  10,  in  10 
and  20,  and  write  the  quotients  1  and  2  above 
the  numbers.  We  then  see  that  16  x  3— 48,  and 
that  4x12=48;  cancel  16  and  3  in  the  divisor, 
and  4  and  12  in  the  dividend  ;  hence,  the  quo- 
tient is  2.  Am 

3.  Divide  the  product  of  126  x  16  x  3,  by  7  x  12. 

ANALYSIS — We  see  that  7  is  a  factor  OPERATION. 

of  126— giving  a    quotient  of  18.     We  1 

cancsl  7,  and   place   18  at  the  right  of 
126.     We  then  cancel  6,  in  12  and  18,     \  * 
and  write   the  quotients  2  and  3.     We          ^ 

then  cancal  the  factor  2,  in  2  and  16,  X 

and   set   down   the  quotients  1   and  8.          Ans.  3x8x3  =  ' 
The    product    of    1x1    is   the    divisor, 
and  the  product  of  3  x  8  x  3  =  72,  the  dividend. 


14:4:  CANCELLATION. 

4.  What  is  the  quotient  of  3x8x9x7x15,  divided  by 
63x24x3x5? 


ANALYSIS.—  The  63  is  cancelled  by  7  x  9  ;  24 
by  3  x  8  ;  3  aiid  5,  by  15  ;  hence,  the  quotient  is  1. 


OPERATION. 
$ 

H 

0 


5.  Divide  the  product  of  6x1x9x11,  by  2x3x7x3 
X21. 

6.  Divide  the  product  of  4 X 14  x  16  x  24,  by  7  x  8x32 
Xl2. 

7.  Divide  the  product  of  5  x  11  x  9  x  7  x  15  x  6,  by  30  x  3 
x21  x3x5. 

8.  Divide  the  product  of  6  x  9  x  8  x  11  x  12  x  5,  by  27  x  2 
x  32  x  3. 

9.  Divide  the  product  of  1  x  6  x  9  x  14  x  15  x  7  x  8,  by  36 
x  126x56x20. 

10.  Divide  the  product  of  18  x  36  x  72  x  144,  by  6  x  6  x  8 
x  9x12x8. 

11.  Divide  the  product  of  4  x  6  x  3  x  5,  by  5  x  9  x  12  x  16. 

12.  Multiply  288  by  16,  and  divide  the  product  by  8  x  9 
x2x2. 

13.  In  a  certain  operation  the  numbers  24,  28,  32,  49,  81, 
are  to  be  multiplied  together  and  the  product  divided  by 
8x4x7x9x6:  what  is  the  result ? 

14.  Multiply  240  by   18   and   divide  the   product  by   6 
times  90. 

15.  Divide  16  x  20  x  8  x  3,  by  30  x  8  x  6. 

16.  How  many  pounds  of  butter  worth  15  cents  a  pound, 
may  be  bought  for  25  pounds  of  tea  at  48  cents  a  pound  ? 

1 7.  How  much  calico  at  25  cents  a  yard  must  be  given 
for  100  yards  of  Irish  sheeting  at  87  cents  a  yard  ? 

18.  How  many  yards  of  cloth  at  46  cents  a  yard  must  be 
given  for  23  bushels  of  rye  at  92  cents  a  bushel  ? 


143.  Give  the  rule  for  the  operation  of  cancellation. 


CANCELLATION.  145 

19.  How  many  bushels  of  oats  at  42  cents  a  bushel  must 
be  given  for  3  boxes  of  raisins  each  containing  26  pounds,  at 
14  cents  a  pound  ? 

20.  A  man  buys  2  pieces  qf  cotton  cloth,  each  containing 
33  yards  at  11  cents  a  yard,  and  pays  for  it  in  butter  at  18 
cents  a  pound  :  how  many  pounds  of  butter  did  IIQ  give  ? 

21.  If  sugar  can  be  bought  for  7  cents  a  pound,  how  many 
bushels  of  oats  at  42  cents  a  bushel  must  I  give  for  56  pounds  ? 

22.  If  wool  is  worth  36  cents  a  pound,  how  many  pounds 
must  be  given  for  27  yards  of  broadcloth  worth  4  dollars  a 
yard? 

23.  If  cotton  cloth  is  worth  9  cents  a  yard,  how  much 
must  be  given  for  3  tons  of  hay  worth  15  dollars  a  ton  ? 

24.  How  much  molasses  at  42  cents  a  gallon  must  be  given 
for  216  pounds  of  sugar  at  7  cents  a  pound? 

25.  Bought  48  yards  of  cloth  at  125  cents  a  yard  :  how 
many  bushels  of  potatoes  are  required  to  pay  for  it  at  150 
cents  a  bushel  ? 

26.  Mr.  Butcher  sold  342  pounds  of  beef  at  6  cents  a 
pound,  and  received  his  pay  in  molasses  at  36  cents  a  gallon : 
how  many  gallons  did  he  receive  ? 

27.  Mr.  Farmer  sold  1263  pounds  of  wool  at  5  cents  a 
pound,  and  took  his  pay  in  cloth  at  421  cents  a  yard  :  how 
many  yards  did  he  take  ? 

28.  How  many  firkins  of  butter,  each  containing  56  pounds, 
at  18  cents  a  pound,  must  be  given  for  3  barrels  of  sugar, 
each  containing  200  pounds,  at  9  cents  a  pound  ? 

29.  How  many  boxes  of  tea,  each  containing  24  pounds, 
worth  5  shillings  a  pound,  must  be  given  for  4  bins  of  wheat, 
each  containing  145  bushels,  at  12  shillings  a  bushel  ? 

30.  A  worked  for  B  8  days,  at  6  shillings  a  day,  for  which 
he  received  12  bushels  of  corn  :  how  much  was  the  corn 
worth  a  bushel  ? 

31.  Bought  15  barrels  of  apples,  each  containing  2  bushels 
at  the  rate  of  3  shillings  a  bushel :  how  many  cheeses,  each 
weighing  30  pounds,  at  1  shilling  a  pound,  will  pay  for  the 
apples  ? 

10 


14:6  COMMON  FRACTIONS. 


COMMON   FRACTIONS. 

144.  The  unit  1  denotes  an  entire  thing,  as  1  apple, 
1  chair,  1  pound  of  tea. 

If  the  unit  1  be  divided  into  two  equal  parts,  each  part 
is  called  one-half. 

If  the  unit  1  be  divided  into  three  equal  parts,  each  part 
is  called  one-third. 

If  the  unit  1  be  divided  into  four  equal  parts,  each  part 
is  called  one-fourth. 

If  the  unit  1  be  divided  into  twelve  equal  parts,  each  part 
is  called  one-twelfth  ;  and  if  it  be  divided  into  any  number 
of  equal  parts,  we  have  a  like  expression  for  each  part. 

The  parts  are  thus  written  : 

is  read,  one-half.  -f  is  read,  one-seventh, 

one-third  |  -     -       one-eighth, 

one-fourth. .  T\T  ~     -      one-tenth. 

-       one-fifth.  T^  -     -       one-fifteenth, 

one-sixth.  ^  -     -       one-fiftieth. 

The  i,  is  an  entire  half;  the  J,  an  entire  third  ;  the  J,  an 
entire  fourth  ;  and  the  same  for  each  of  the  other  equal  parts  : 
hence,  each  equal  part  is  an  entire  thing,  and  is  called  a  frac- 
tional unit. 

The  unit  1 ,  or  whole  thing  which  is  divided,  is  called  the 
unit  of  the  fraction. 

NOTE. — In  every  fraction  let  the  pupil  distinguish  carefully 
between  the  unit  of  the  fraction  and  the  fractional  unit.  The  first 
is  the  whole  thing  from  which  the  fraction  is  derived  ;  the  second, 
one  of  the  equal  parts  into  which  that  thing  is  divided. 

145.  Each  fractional  unit  may  become  the  base  of  a  col- 
lection of  fractional  units :  thus,  suppose  it  were  required  to 
express  2  of  each  of  the  fractional  units  :  we  should  then  write 

144.  What  is  a  unit  ?  What  is  each  part  called  when  the  unit  1  is 
divided  into  two  equal  parts  ?  When  it  is  divided  into  3  ?  Into  4?  Into 
5?  Into  12? 

How  may  the  one-half  be  regarded  ?  The  one-third  ?  The  one-fourth  ? 
What  is  each  part  called  ? 

What  is  the  unit  of  a  fraction  ?  What  is  a  fractional  unit  ?  How  do 
you  distinguish  between  the  one  and  the  otlu-r  ? 


COMMON   FRACTIONS. 

which  is  read     2  halves  =  J  x  2 
"     "     "         2  thirds  =Jx2 
2fourths=Jx2 
2  fifths    =£x2 
&c.,     &c.,     &c.f     &c. 

If  it  were  required  to  express  3  of  each  of  the  fractional 
units,  we  should  write  % 

-|       which  is  read  3  halves  =^  x  3 

f          "     «     «  3  thirds  =4x3 

£           "     "     "  3  fourths  =1x3 

J           "     "     "  3  fifths    =1x3 

&c.,     &c.,     &c.,  &c.  ;  hence, 

A  FRACTION  is  one  of  the  equal  parts  of  the  unit  1,  or  a 
collection  of  such  equal  parts. 

Fractions  are  expressed  by  two  numbers,  the  one  written 
above  the  other,  with  a  line  between  them.  The  lower  num- 
ber is  called  the  denominator,  and  the  upper  number  the 
numerator. 

The  denominator  denotes  the  number  of  equal  parts  into 
which  the  unit  is  divided  ;  and  hence,  determines  the  value 
of  the  fractional  unit.  Thus,  if  the  denominator  is  2,  the 
fractional  unit  is  one-half;  if  it  is  3,  the  fractional  unit  is  one- 
third  ;  if  it  is  4,  the  fractional  unit  is  one-fourth,  &c.,  &c. 

The  numerator  denotes  the  number  of  fractional  units  taken. 
Thus,  -f  denotes  that  the  fractional  unit  is  ^,  and  that  3  such 
units  are  taken  ;  and  similarly  for  other  fractions. 

In  the  fraction  f ,  the  base  of  the  collection  of  fractional 
units  is  £,  but  this  is  not  the  primary  base.  For,  •£  is  one- 
fifth  of  the  unit  1  ;  hence,  the  primary  base  of  every  fraction 
is  the  unit  1. 

145.  May  a  fractional  unit  become  the  base  of  a  collection  ?  What  is 
a  fraction  ?  How  are  fractions  expressed  ?  What  is  the  lower  number 
called  ?  What  is  the  upper  number  called  ?  What  does  the  denomina- 
tor denote?  What  does  the  numerator  denote?  In  the  fraction 
3  fifths,  what  is  the  fractional  base  ?  What  is  the  primary  base  ?  What 
is  the  primary  base  of  every  fraction  ? 


148  COMMON   FRACTIONS. 

146.  If  we  take  other  units  1,  each  of  the  same  kind,  and 
divide  each  into  equal  parts,  such  parts  may  be  expressed 
in  the  same  collection  with  the  parts  of  the  first :  thus, 

f  is  read  3  halves. 

I  "     "  ?  fourths. 

i/-  "     "  16  fifths. 

V  "  .  *'  18  sixths. 

•2j&-  25  sevenths. 

147.  A  whole  number  may  be  expressed  fractionally  by 
writing  1  below  it  for  a  denominator.     Thus, 

3  may  be  written  -f-  and  is  read,  3  ones. 
5--  -  •{•---  5  ones. 
6  -  -  f  -  -  -  6  ones. 

8    -      -        -       -f-    -    -     -      8  ones. 

But  3  ones  are  equal  to  3,  5  ones  to  5,  6  ones  to  6,  and 
8  ones  to  8  ;  hence,  the  value  of  a  number  is  not  changed  by 
placing  1  under  it  for  a  denominator. 

148.  If  the  numerator  of  a  fraction  be  divided  by  its  de- 
nominator, the  integral  part  of  the  quotient  will  express  the 
number  of  entire  units  used  in  forming  the  fraction  ;  and  the 
remainder  will    show  how  many  fractional   units  are  over. 
Tims,  JyL  are  equal  to  3  and  2  thirds,  and  is  written  -V-— 3I : 
hence, 

A  fraction  has  the  same  form  as  an  unexecuted  division. 

From  what  has  been  said,  we  conclude  that, 

1st.  A  fraction  is  one  or  more  of  the  equal  parts  of  the 
unit  1. 

2d.  The  denominator  shows  into  how  many  equal  parts 
the  unit  is  divided,  and  hence  indicates  the  value  of  the 
fractional  unit : 

146.  If  a  second  unit  be  divided  into  equal  parts,  may  the  parts  be 
expressed  with  those  of  the  first?    How  many  units  have  been  divided 
to  obtain  6  thirds  ?    To  obtain  9  halves  ?    12  fourths  ? 

147.  How  may  a  whole  number  be  expressed    fractionally?     Does 
this  change  the  value  of  the  number? 

148.  If  the  numerator  be  divided  by  the  denominator,  what  docs  the 
quotient  show?    What  does  the  remainder  show?    What  form  has  a 
fraction  ?    What  are  the  seven  principles  which  follow  ? 


COMMON  FRACTIONS.  149 

3d.  The  numerator  shows  how  many  fractional  units  are 
taken : 

4th.  The  value  of  every  fraction  is  equal  to  the  quotient 
arising  from  dividing  the  numerator  by  the  denominator. 

5th.  When  the  numerator  is  less  than  the  denominator, 
the  value  of  the  fraction  is  less  than  1. 

6th.  When  the  numerator  is  equal  to  the  denominator, 
the  value  of  the  fraction  is  equal  to  1. 

7th.  When  the  numerator  is  greater  than  the  denomina- 
tor, the  value  of  the  fraction  is  greater  than  1 

EXAMPLES    IN    WRITING   AND    READING    FRACTIONS. 

1.  Read  the  following  fractions  ; 

T5u,  f ,  ¥,  T7o,  f ,   59o,  T¥T. 

What  is  the  unit  of  the  fraction,  and  what  the  fractional  unit, 
in  each  example  ?  How  many  fractional  units  are  taken  in  each? 

2.  Write  12  of  the  17  equal  parts  of  1. 

3.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit 
one-twentieth,  express  6  fractional  units.     Express  12,  18, 
16,  30,  fractional  units. 

4.  If  the  fractional  unit  is  one  36th,  express  32  fractional 
units  ;  also,  35,  38,  54,  6,  8. 

5.  If  the  fractional  unit  is  one-fortieth,  express  9  fractional 
units  ;  also,  16,  25,  69,  75. 

DEFINITIONS. 

149.  A  PROPER  FRACTION  is  one  whose  numerator  is  less 
than  the  denominator. 

Tue  following  are  proper  fractions : 

i  i   i   I  f  J,  A,  t,  *• 

150.  An  IMPROPER   FRACTION  is  one  whose   numerator  is 
equal  to,  or  exceeds  the  denominator. 

NOTE. — Such  a .  fraction  is  called  improper  because  its  value 
equals  or  exceeds  1. 

149.  What  is  a  proper  fraction  ?    Give  examples. 

150.  What  is  an  improper  fraction  ?    Why  improper  ?    Give  exam- 
ples. 


150  PROPOSITIONS   IN 

The  following  are  improper  fractions  : 

4,  4,  4,  4,  f ,  4,  ¥,  •¥,  V- 

151.  A  SIMPLE  FRACTION  is  one  whose  numerator  and  de- 
nominator are  both  whole  numbers. 

NOTE. — A  simple  fraction  may  be  either  proper  or  improper. 
The  following  are  simple  fractions  : 

i  f ,  *,  f ,  4,  4,  4.  *• 

152.  A  COMPOUND  FRACTION  is  a  fraction  of  a  fraction,  or 
several  fractions  connected  by  the  word  of,  or  x  . 

The  following  are  compound  fractions  : 

Jofi    iofiofj,  £x3,  ixJx-4. 

153.  A  MIXED  NUMBER  is  made  up  of  a  whole  number  and 
a  fraction. 

The  following  are  mixed  numbers  : 

3i,     41,     6f,     54,     6|,     3f 

154.  A  COMPLEX  FRACTION  is  one  whose  numerator  or  de- 
nominator is  fractional ;  or,  in  which  both  are  fractional. 

The  following  are  complex  fractions : 

j  2  f  45t 

5  191'  *'  69V 

155.  The  numerator  and  .denominator  of  a  fraction,  taken 
together,  are  called  the  terms  of  the  fraction  :  hence,  every 
fraction  has  two  terms. 

FUNDAMENTAL    PROPOSITIONS. 

156.  By  multiplying  the  unit  1,  we  form  all  the  whole 
numbers, 

151.  What  is  a  simple  fraction  ?    Give  examples.    May  it  be  proper 
or  improper  ? 
153.  What  is  a  compound  fraction  ?    Give  examples. 

153.  What  is  a  mixed  number  ?    Give  examples. 

154.  What  is  a  complex  fraction  ?    Give  examples. 

155.  How  many  terms  has  every  fraction  ?     What  are  they  ? 

156.  How  may  all  the  whole  numbers  be  formed?    How  may  the 
fractional  units  be  formed  ?    How  many  times  is  one-half  less  than  1  ? 
How  many  times  is  any  fractional  unit  less  than  1  ? 


COMMON   FRACTIONS.  151 

2,    3,    4,    5,    6,    1,    8,    9,    10,    &c.  ; 
and  by  dividing  the  unit  1  by  these  numbers  we  form  all  the 
fractional  units, 

i'   4'   I*  i>  i'  I'  i>  I'  A»  &c- 

Now,  since  in  1  unit  there  are  2  halves,  3  thirds,  4 
fourths,  5  fifths,  6  sixths,  &c.,  it  follows  that  the  fractional 
unit  becomes  less  as  the  denominators  are  increased :  hence, 

The  fractional  unit  is  such  a  part  of  I,  as  I  is  of  the 
denominator  of  the  fraction. 

Thus,  J  is  such  a  part  of  1,  as  1  is  of  2  ;  J  is  such  a  part  of 
1,  as  1  is  of  3-;  J  is  such  a  part  of  1  as  1  is  of  4,  &c.  &c. 
157.  Let  it  be  required  to  multiply  £  by  3. 

ANALYSIS. — In  f  there  are  5  fractional  OPERATION-. 

units,  each  of  which  is  ^,  and  these  are  to         4  x  3^-5-vp-— J^A 
be  taken  3  times.     But  5  things  taken  3 

times,  gives  15  things  of  the  same  kind  ;  that  is,  15  sixths  :  hence, 
the  product  is  3  times  as  great  as  the  multiplicand :  therefore,  we 
have 

PROPOSITION  I. — If  the  numerator  of  a  fraction  be  multi- 
plied by  any  number,  the  value  of  the  fraction  will  be  in- 
creased as  many  times  as  there  are  units  in  the  multiplier. 


4.  Multiply  TV  by  14. 

5.  Multiply  %  by  20. 

6.  Multiply  Jj&z-  by  25 


EXAMPLES. 

1.  Multiply  -3  by  8. 

2.  Multiply  I  by  5. 

3.  Multiply  \  by  9. 

158.  Let  it  be  required  to  multiply  £  by  3. 

ANALYSIS. — In  £  there  are  4  fractional  OPERATION. 

units,  each  of  which  is  £.    If  we  divide          4- X  3 —    4    —  ±. 
the  denominator  by  3,  we  change  the  frac-  6~3 

tional  unit  to  \,  which  is  3  times  as  great  as  £,  since  the  first  is 
contained  in  1,  2  times,  and  the  second  6  times.  If  we  take  this 
fractional  unit  4  times,  the  result  £,  is  3  times  as  great  as  $: 
therefore,  we  have 

PROPOSITION  II. — If  the  denominator  of  a  fraction  be  divi- 
ded by  any  number,  the  value  of  the  fraction  will  be  in- 
creased as  many  times  as  there  are  units  in  that  number. 

157.  What  is  proved  in  Proposition  I.  ? 


152  PROPOSITIONS  IN 


EXAMPLES. 


4.  Multiply  H  by  2,  4,  6. 

5.  Multiply  ££  by  2,  6,  7. 

6.  Multiply  $fo  by  5,  10. 


1.  Multiply  |  by  2,  by  4. 

2.  Multiply  Jf  by  2,  4,  8. 

3.  Multiply  ^  by  2,  4,  6. 

159.  Let  it  be  required  to  divide  fa  by  3. 

ANALYSIS. — In  -ft,  there  are  9  fractional  OPERATION. 

units,  each  of  which  is  -,1,-,  and  these  are  s '  -f-3—  9-3—  -3 

to  be  divided  by  3.     But  9  things,  divided  1 1 

by  3,  gives  3  things  of  the  same  kind  for  a  quotient ;  hence,  the 
quotient  is  3  elevenths,  a  number  one-third  as  great  as  -ft ;  hence, 
we  have 

PROPOSITION  III. — If  the  numerator  of  a  fraction  be  divi- 
ded by  any  number,  the  value  of  the  fraction  will  be  dimin- 
ished as  many  times  as  there  are  units  in  the  divisor. 


EXAMPLES. 


1.  Divide  ff  by  2,  by  7 

2.  Divide  $J  by  56. 


3.  Divide  f££  by  25,  by  8. 

4.  Divide  ff£  by  8,  16,  10. 

1GO.  Let  it  be  required  to  divide  fa  by  3. 

ANALYSIS. — In  -ft-,  there  are  9  fractional  OPERATION. 

units,  each  of  which  is  -ft-.     Now.  if  we        $  -^-3=^ *—-£r. 

multiply  the  denominator  by  3  it  becomes 

33,  and  the  fractional  unit  becomes  -^-j,  which  is  only  ^  of  -,1,-,  be- 
cause 33  is  3  times  as  great  as  11.  If  we  take  this  fractional 
unit  9  times,  the  result,  -£,-,  is  exactly  ^  of  -ft :  hence,  we 
have 

PROPOSITION  IY. — If  the  denominator  of  a  fraction  be 
multiplied  by  any  number,  the  value  of  the  fraction  will  be 
diminished  as  many  times  as  there  are  units  in  that  number. 


EXAMPLES. 


1.  Divide  \  by  2. 

2.  Divide  £  by  1. 

3.  Divide  -^  by  4. 


4.  Divide  f£  by  8. 

5.  Divide  fj-  by  17. 

6.  Divide  TV%  by  45. 


158.  What  is  proved  in  proposition  II.  ? 

159.  What  is  proved  in  proposition  III.  ? 
100.  What  is  proved  in  proposition  IV.  ? 


COMMON   FRACTIONS.  153 

161.  Let  it  be  required  to  multiply  both  terms  of  the  frac- 
tion f  by  4. 

ANALYSIS.  —  In  f,  the  fractional  unit  is  £,  and  it  OPERATION. 
is  taken  3  times.  By  multiplying  the  denominator  ?lf  —-JL2.. 

by  4,  the  fractional  unit  becomes  ^7,  the  value  of  5x4~~^o 

which  is  ^  times  as  as  great  as  i.  By  multiplying  the  numerator 
by  4,  we  increase  the  number  of  fractional  units  taken,  4  times, 
that  is,  we  increase  the  number  just  as  many  times  as  we  decrease 
the  value  ;  hence,  the  value  of  the  fraction  is  not  changed  ;  there- 
fore, we  have 

PROPOSITION  Y.  —  If  both  terms  of  a  fraction  be  multiplied 
by  the  same  number,  the  value  of  the  fraction  will  not  be 
changed. 

EXAMPLES. 

1.  Multiply  the  numerator  and  denominator  of  -f-  by  7  : 
this  gires  ^H—M. 

7X  7  —  49 

2.  Multiply  the  numerator  and  denominator  of  -fa  by  3,  by 
4,  by  5,  by  6,  by  9. 

3.  Multiply  each  term  of  £|  by  2,  by  3,  by  4,  by  5,  by  6. 

162.  Let  it  be  required  to  divide  the  numerator  and  de- 
nominator of  T63-  by  3. 

ANALYSIS.  —  In  -rV,  the  fractional  unit  is  -fa,  and  OPERATION. 
is  taken  6  times.  By  dividing  the  denominator  6  -r-3__2 
by  3,  the  fractional  unit  becomes  i,  the  value  of  T^_^o  —  7"* 
which  is  3  times  as  great  as  -fa.  By  dividing  the 
numerator  by  3,  we  diminish  the  number  of  fractional  units  taken 
3  times  :  that  is,  we  diminish  the  number  just  as  many  times  as  we 
increase  the  value  :  hence,  the  value  of  the  fraction  is  not  changed  : 
therefore  we  have 

PROPOSITION  YI.  —  If  both  terms  of  a  fraction  be  divided 
by  the  same  number,  the  value  of  the  fraction  will  not  be 
changed. 

EXAMPLES. 

1.  Divide  both  terms  of  the  fraction  ^  by  2  :  this  gives 
±=     Ans. 


161.  What  is  proved  hy  proposition  V.  ? 

162.  What  is  proved  by  proposition  VI.  ? 


154  REDUCTION   OF 

2.  Divide  both  terms  by  8  :  this  gives  ^  ±f = J. 

3.  Divide  both  terms  of  the  fraction  -j3^-  by  2,  by  4,  by  8, 
by  16. 

4.  Divide  both  terms  of  the  fraction  T^j  by  2,  by  3,  by  4, 
by  5,  by  6,  by  10,  by  12. 

REDUCTION  OF  FRACTIONS. 

163.  REDUCTION  OF  FRACTIONS  is  the  operation  of  changing 
the  fractional  unit  without  altering  the  value  of  the  fraction. 

A  fraction  is  in  its  lowest  terms,  when  the  numerator  and 
denominator  have  no  common  factor. 

CASE  i. 

164.   To  reduce  a  fraction  to  its  lowest  terms. 
1.  Reduce  TW  to  its  lowest  terms. 

ANALYSIS. — By  inspection,  it  is  seen  that  5 

is  a   common   factor  of    the   numerator  and        IST  OPERATION. 
denominator.     Dividing   by   it,    we   have   if.  5)T7-*yr  — 4-i. 

We  then  see  that  7  is  a  common  factor  of  14 
and   35:  dividing   by    it,  we   have   £.      Now,  fr\i4_2 

there  is  no  common  factor  to  2  and  5  :  hence,  'So  —  t' 

§  is  in  its  lowest  terms. 

The  greatest  common  divisor  of  70  and  175         2D  OPERATION. 
is  35,  (Art.  136);  if  we  divide  both  terms  of          35)  TJ^  — .2. 
the  fraction  by  it,  we  obtain  £.     The  value  of 
the  fraction  is  not  changed  in  either  operation,  since  the  numera- 
tor and  denominator  are  both  divided  by  the  same  number  (Art. 
162):  hence,  the  following 

RULE. — Divide  the  numerator  mid  denominator  by  any 
number  that  will  divide  them  both  without  a  remainder,  and 
divide  the  quotient,  in  the  same  manner  until  they  have  no 
common  factor. 

Or :  Divide  the  numerator  and  denominator  by  their  great- 
est common  divisor. 

163.  What  is  reduction  of  fractions  ?    When  is  a  fraction  in  its  lowest 
terms  ? 

164.  How  do  you  reduce  a  fraction  to  its  lowest  terms  ? 


COMMON   FRACTIONS.  155 

EXAMPLES. 

Reduce  the  following  fractions  to  their  lowest  terms. 


1.  Reduce  -ff. 

2.  Reduce  ff. 

3.  Reduce  f£. 

4.  Reduce 

5.  Reduce 

6.  Reduce 
V.  Reduce 
8.  Reduce 


9.  Reduce 

10.  Reduce 

11.  Reduce 

12.  Reduce 

13.  Reduce 

14.  Reduce 

15.  Reduce 

16.  Reduce 


CASE    II. 

165.   To  reduce  an  improper  fraction  to  its  eouivalent 
whole  or  mixed  number. 

1.  In  $/•  how  many  entire  units  ? 

ANALYSIS. — Since  there  are  8  eighths  in  1  unit,  OPERATION. 
in  *£•  there  are  as  many   units  as  8  is  contain-  8)59 

ed  times  in  59,  which  is  7|  times.  — =-£- 

Hence,  the  following 

RULE. — Divide  the  numerator  by  the  denominator,  and  the 
result  ivill  be  the  whole  or  mixed  number. 

EXAMPLES. 

1.  Reduce  &£•  and  fy  to  their  equivalent  whole  or  mixed 
numbers. 

OPERATION.  OPERATION. 

4)84  9)67 

2.  Reduce  sg.  to  a  whole  or  mixed  number. 

3.  In  I?9-  yards  of  cloth,  how  many  yards  ? 

4.  In  -^L  of  bushels,  how  many  bushels  ? 

165.  How  do  you  reduce  an  improper  fraction  to  a  whole  or  mixed 
number  ? 


156  REDUCTION    OF 

5.  If  I  give  I  of  an  apple  to  each  one  of  15  children,  how 
many  apples  do  I  give  ? 

6.  Reduce  ffj,  3ff£,  JtfffiL,  *fj£#f.t  to  their  whole  or 
mixed  numbers. 

7.  If  I  distribute  878  quarter-apples  among  a  number  of 
boys,  how  many  whole  apples  do  I  use  ? 

8.  Reduce  %5T8^,  \W,  WsWeS  to  tneir  whole  or  mixed 
numbers. 

9.  Reduce  JLt^ffi^  J^\^a,  £2^p£}  to  their  whole 
or  mixed  numbers. 

CASE    III. 

160.  To  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 

1.  Reduce  4f-  to  its  equivalent  improper  fraction. 

ANALYSis.-Since  '  in    any    number  OPERATION. 

there  are   5    times   as   many   fifths   as  A  ..  r On  Gp^n 

units,  in  4  there  will  be  5  times  4  fifths, 

or  20  fifths,  to  which  add  4  fifths,  and  add         4  fifths. 

we  have  24  fifths.  gives  •%*•  =  24  fifths. 

Hence,  the  following 

RULE. — Multiply  the  whole  number  by  the  denominator  of 
the  fraction :  to  the  product  add  the  numerator,  and  place  the 
sum  over  the  given  denominator. 

EXAMPLES. 

1.  Reduce  47f  to  its  equivalent  fraction. 

2.  In  It  £  yards,  how  many  eighths  of  a  yard? 

3.  In  42  -/^  rods,  how  many  twentieths  of  a  rod  ? 

4.  Reduce  625-^-  to  an  improper  fraction. 

5.  How  many  112ths  in  205T4T%  ? 

6.  In  84^£  days,  how  many  twenty-fourths  of  a  day  ? 

7.  In  15J$|  years,  how  many  365ths  of  a  year  ? 

8.  Reduce  916££-{}  to  an  improper  fraction. 

9.  Reduce  25T%-,  156f^,  to  their  equivalent  fractions. 

100.  How  do  you  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 


COMMON   FRACTIONS.  157 

CASE    IV. 

167.  To  reduce  a  whole  number  to  a  fraction  having  a 
given  denominator. 

1.  Reduce  6  to  a  fraction  whose  denominator  shall  be  4. 

ANALYSIS. — Since  in  1  unit  there  are  4  fourths,  OPERATION. 
it  follows  that  in  6  units  there  are  6  times  4  fourths,  6x4  —  24. 
or  24  fourths:  therefore,  6=Y  •  hence,  .gjt 

RULE. — Multiply  the  whole  number  and  denominator 
together,  and  write  the  product  over  the  required  denomi- 
nator. 

EXAMPLES. 

1.  Reduce  12  to  a  fraction  whose  denominator  shall  be  9. 
2  Reduce  46  to  a  fraction  whose  denominator  shall  be  15. 


3.  Change  26  to  7ths. 

4.  Change  178  to  40ths. 

5.  Reduce  240  to  IHths. 


6.  Change  $54  to  quarters. 

7.  Change  96?/<^.  to  quarters. 

8.  Change  426/6.  to  16ths. 


CASE   V. 

168.   To  reduce  a  compound  fraction  to  a  simple  one. 
1.  What  is  the  value  of  £  of  f? 

ANALYSIS. — Three-fourths  of  f  is  3  times  1  fourth  OPERATION. 
of  $ ;  1  fourth  of  f  is  &  (Art.  160) ;  3  fourths  of  f  is  3x5  15 
3  times  &,  or  if :  therefore,  f  of  $=i£ :  hence,  -= =— 

4x7      2o 

RULE. — Multiply  the  numerators  together  for  a  new 
numerator,  and  the  denominators  together  for  a  new  de- 
nominator. 

NOTE. — If  there  are  mixed  numbers,  reduce  them  to  their  equiv- 
alent improper  fractions. 

EXAMPLES. 

P*educe  the  following  fractions  to  simple  ones. 


1.  Reduce  J  of  J  of  f. 

2.  Reduce  £  of  £  of  f. 

3.  Reduce  f  of  f  o 


4.  Reduce  2J  of  6J  of  7. 

5.  Reduce  5  of  \  of  |  of  6. 

6.  Reduce  6^  of  7}  of  6ff. 


158  REDUCTION    OF 

•  METHOD    BY    CANCELLING. 

169.  The  work  may  often  be  abridged  by  cancelling  com- 
mon factors  in  the  numerator  and  denominator  (Art.  143). 

In  every  operation  in  fractions,  let  this  be  done  whenever 
it  is  possible. 

EXAMPLES. 

1.  Reduce  f  of  f  of  -f  to  a  simple  fraction. 

5 


Here, 


7  |  5=f 


NOTE.  —  The  divisors   are  always  written  on  the  left   of  the 
vertical  line,  and  the  dividends  on  the  right. 


2 


2.  Reduce  £  of  f  of  T^  to  its  simplest  terms. 

!  *    *    2  *  * 

-rr  V         V  r  ^ 

TT  ^-«»o  __-  NX    —  V ^^  —T-  . »  rv-t* 

xi ere,         »i  A  A  x  vet  —  F;  UI »  R 


5  |  2=2. 

NOTE.  —  Besides  cancelling  the  like  factors  8  and  8,  and  9  and  9> 
we  also  cancel  the  factor  3,  common  to  15  and  6,  and  write  ovei 
them,  and  at  the  left  and  right,  the  quotients  5  and  2. 

3.  Reduce  |  of  -f  of  •§•  of  -fife  of  T5^  to  its  simplest  terms. 

4.  Reduce  -f-fc  of  T\  of  T%  of  f  to  its  simplest  terms. 

5.  Reduce  3|-  of  f  of  ^  of  49  to  its  simplest  terms. 


CASE    TI. 

170.  To  reduce  fractions  of  different   denominators  to 
fractions  having  a  common  denominator. 

1.  Reduce  \,  %  and  4  to  a  common  denominator. 

167.  How  do  you  reduce  a  whole  number  to   a  fraction  having  a 
given  denominator? 

168.  How  do  you  reduce  a  compound  fraction  to  a  simple  one  ? 

169.  How  is  the  reduction  of   compound  fractions  to  simple  ones 
abridged  by  cancellation. 


COMMON  FRACTIONS.  159 

ANALYSIS.— If  both  terms  of  the  OPERATION. 

first    fraction    be    multiplied  by  15,         1x3x5=15   1st  num. 
the  product  of  the  other  denomina-        7x2x5  =  70  2d  num. 
tors,    it    will    become    £ft.     If  both        iv3v9  —  24-  3r1   nnm 
terms  of  the  second  fraction  be  mul- 
tiplied by  10,    the    product  of  the        2x3x5  =  dO  clenom. 
other  denominators,    it   will  become  £$.    If  both  terms  of  the 
third  be  multiplied  by  6,  the  product  of  the  other  denominators, 
it  will  become  f  £.     In  each  case,  we  have  multiplied  both  terms 
of  the  fraction   by  the  same   number ;  hence,  the  value  has  not 
been  altered  (Art.  161) :  hence,  the  following 

RULE. — Eeduce  to  simple  fractions  when  necessary  ;  then 
multiply  the  numerator  of  each  fraction  by  all  the  denomi- 
nators except  its  own,  for  the  new  numerators,  and  all  the 
denominators  together  for  a  common  denominator. 

NOTE. — When  the  numbers  are  small  the  work  may  be  per- 
formed mentally.  Thus, 

i-  \f  *=»•  «•  «• 

EXAMPLES. 

Reduce  the  following  fractions  to  common  denominators. 


1.  Reduce  f,  f,  and  -$-. 

2.  Reduce  f ,  -f^-,  and  f . 

3.  Reduce  4f-,  |,  and  $. 

4.  Reduce  2J,  and  J  of -f. 

5.  Reduce  5  J,f  of  J,  and  4. 


6.  Reduce  3£  of  J  and  f  . 

7.  Reduce  £,Y/,  and  37. 


8.  Reduce  4,  fj,  and  ££. 

9.  Reduce  7J,  ffr,  6J. 

10.  Reduce  4£,  8|,  and  2|. 


NOTE. — We  may  often  shorten  the  work  by  multiplying  the  nu- 
merator and  denominator  of  each  fraction  by  such  a  number  as 
will  make  the  denominators  the  same  in  all. 

10.  Reduce  J  and  J  to  a  common  denominator. 

OPERATION. 

ANALYSIS.— Multiply  both  terms  of  the  first  by  1=4 

3,  and  both  terms  of  the  second  by  2.  l—s. 

3        <» 


11.  Reduce  £  and  J. 

12.  Reduce  £,  ^,  and  }. 

13.  Reduce  -. 


14.  Reduce  f ,  3£,  and  |. 

15.  Reduce  6^,  9J,and5. 

16.  Reduce  7f,f,  J,  and£. 


170.  How  do  you  reduce  fractions  of  different  denominators  to  frac- 
tions having  a  common  denominator  ?  When  the  numbers  are  small, 
how  may  the  work  be  performed  ? 


160  REDUCTION    OF 

CASE   VII. 

171.  To  reduce  fractions  to  their  least  common  denominator. 

The  least  common  denominator  is  the  number  which  con- 
tains only  the  prime  factors  of  the  denominators. 

1.  Reduce  J,  f ,  and  |,  to  their  least  common  denominator. 

OPERATION. 

(12-=-3)xl  =  4  1st  Numerator.         3)3     .     6     .     4 
(12-^-6)  x  5  =  10  2d  "  2)1     .     2     .~4~ 

(12-T-4)x3=  9  3d  "  1.1.2 

3x2x2  =  1 2,  least  com.  denom. 

Therefore,  the  fractions  J,  f,  and  f,  reduced  to  their  least 
common  denominator,  are  T%,  -ff,  and  T\. 
Hence,  the  following 

RULE, — I.  Find  the  least  common  multiple  of  the  denomi- 
nators (Art.  140),  which  will  be  the  least  common  denominator 
of  the  fractions. 

II.  Divide  the  least  common  denominator  by  the  denomina- 
tors of  the  given  fractions  separately,  and  multiply  the  nume- 
rators by  the  corresponding  quotients,  and  place  the  products 
over  the  least  common  denominator. 

NOTES. — 1.  Before  beginning  the  operation,  reduce  every  frac- 
tion to  a  simple  fraction  and  to  its  lowest  terms. 

2.  The    expressions,    (12-r-3)xl,    (12-7-6)  x  5,    (12-f-4)x3,    indi- 
cate that  the  quotients  are  to  be  multiplied  by  1,  5,  and  3. 

EXAMPLES. 

Reduce  the  following  fractions  to  their  least  common 
denominator. 

2.  Reduce  f ,  f ,  T3T. 

3.  Reduce  14f,  6-f,  5J. 

4.  Reduce  -^  -fa,  f . 

5.  Reduce  -flfr,  ^,  f. 

6.  Reduce  ££,  3^,  4. 


1.  Reduce  3|, 

8.  Reduce  J,  §,  j,  and  £. 

9.  Reduce  2J  of  £,  3}  of  2. 

10.  Reduce  -f,  f ,  £ ,  and  TV 

11.  Reduce  J,  f,  f,  I  «. 


171.  Wliat  is  the  least  common  denominator  of  several  fractions? 
How  do  you  reduce  fractions  to  their  least  common  denominator  V 


COMMON   FRACTIONS. 


161 


OPERATION. 


OPERATION. 


ADDITION  OF  FRACTIONS. 

172.  Addition  of  Fractions  is  the  operation  of  finding  the 
number  of  fractional  units  in  two  or  more  fractions. 

1.  What  is  the  sum  of  J,  f ,  and  f  ? 

ANALYSIS. — The  fractional  unit  is  the  same 
in  each  fraction,  viz. :  ^  ;  but  the  numerators 
show  how  many  such  units  are  taken  (Art.  148) ; 
hence,  the  sum  of  the  numerators  written  over 
tJie  common  denominator,  expresses  the  sum  of  Ans.  f =4£. 
the  fractions. 

2.  What  is  the  sum  of  J  and  f  ? 

ANALYSIS. — In  the  first,  the  fractional  unit 
is  £,  in  the  second  it  is  ^.  These  unite,  not 
being  of  the  same  kind,  cannot  be  expressed  in 
the  same  collection.  But  the  £=f,  and  f =$, 
in  each  of  which  the  unit  is  £:  hence,  their 
sum  is  ^=1^. 

NOTE. — Only  units  of  the  same  kind,  whether  fractional  or  inte- 
gral, can  be  expressed  in  the  same  collection, 

From  the  above  analysis,  we  have  the  following 

RULE. — I.  When  the  fractions  have  the  same  denominator, 
add  the  numerators,  and  place  the  sum  over  the  common  deno- 
minator. 

II.  When  they  have  not  the  same  denominator,  reduce  them 
to  a  common  denominator,  and  then  add  as  before. 

NOTE. — After  the  addition  is  performed,  reduce  every  result  to 
its  lowest  terms. 


*-* 


EXAMPLES. 


1.  Add  J,  f ,  f ,  and  f . 

2.  Add  |,  f ,  and  f 

3.  Addf,±  f,^,an 

4.  Add  t^,^,  an 

5.  Add  f ,  .ft,  and  ft. 

6.  Add  i,  |,  f,  and  ft. 

7.  Add  |,  I  fc  and  ft. 


8.  Add  |,  I  i  and  -ft. 

9.  Add  9,  |,  TV,  f ,  and  f . 

10.  Add  J,  f ,  f ,  1,  and  f 

11.  Add  fV,  f ,  A,  and  f . 

12.  Add  |,  f ,  and  f. 

13.  Add  TV,  f ,  f,  and  f . 

14.  Add  -!%,  f,  f,  and  ^. 


162  SUBTRACTION  OF 

15.  What  is  the  sum  of  19},  6§,  and  4|? 

OPERATION. 

Whole  numbers.  Fractions. 

19  +  6+4=29^          ^  *+§+*=«*= 

17o.  NOTE. — When  there   are  mixed  numbers,  add  the  uhole, 
numbers  and  fractions  separately,  and  then  add  their  sums. 

Find  the  sums  of  the  following  fractions  : 

16.  Add  3J,  7y%,  12f,  1?.         20.  Add  900TV,  450£, 

17.  Add  16,  9|,  25£,  T£.  21.  AddJofT3TofT£to 

18.  Add  |  of  |,  4.  of  9,  14TV     22.  Add  17|  to  f  of  27$. 

19.  Add  2T8T,  6£,  and  12-if.      23.  Add  $,  7J,  and  8|. 

24.  What  is  the  sum  cf  |  of  12§  of  7|,  and  $  of  25  ? 

25.  What  is  the  sum  of  -fa  of  9f  and  -^  of  328f  ? 

174.  1.  What  is  the  sum  of -J-  and  £? 

NOTE. — If  each  of  the  two  fractions  has  OPERATION. 

1  for  a  numerator,  the  sum  of  the  frac-  A- +1  —  c  +  5  —  il 
tions  will  be  equal  to  the  sum  of  their  _5  +  G_ 

denominators  divided  by  their  product.  ~5  «  ¥  —  j^~G"  —  ao' 

2.  What  is  the  sum  of  |  and  ^-  ?  of  £  andTV  ? 

3.  What  is  the  sum  of  -f  and  -fa  ?  of  T\j-  and  y1^-  ?  of  T^ 
andi? 

4.  What  is  the  sum  of  J   and  yV?   °f  1  and  £?   of  J 
and  yV  ? 

SUBTRACTION  OF  FRACTIONS. 

175.  SUBTRACTION  of  Fractions  is  the  operation  of  finding 
the  difference  between  two  fractions. 


173.  What  is  addition  of  fractions  ?  When  the  fractional  unit  is  the 
same,  what  is  the  sum  of  the  fractions  ?  What  units  may  be  expressed 
in  the  same  collection  ?  What  is  the  rule  for  the  addition  of  fractions  ? 

173.  When  there  are  mixed  numbers,  how  do  you  add  ? 

174.  When  two  fractions  have  1  for  a  numerator,  what  is  their  sum 
equal  to  ? 

175.  What  is  subtraction  of  fractions  ? 


COMMON  FBACTIONS. 


163 


1.  What  is  the  difference  between  £•  and  f  ? 

ANALYSIS.  —  In  this  example  the  fractional  unit 
is  i  :  there  are  5  such  units  in  the  minuend  and 
3  in  the  subtrahend  :  their  difference  is  2  eighths  ; 
therefore,  2  is  written  over  the  common  denomi- 
nator 8. 


2.  From  J^.  take  -i 

3.  From  -|  take  f . 


4.  From 

5.  From 


OPERATION. 


take 
take 


OPERATION. 


i  —  .  4 

jj,  __  y*  _  g  _ 
ttr  ~T1F  —  TT  — 


6.  What  is  the  difference  between     and 


ANALYSIS.  —  Reduce  both  to  the  same  frac- 
tional  unit  -^  :  then,  there  are  10  sucli  units 
in  the  minuend  and  4  in  the  subtrahend: 
hence,  the  difference  is  6  twelfths. 


From  the  above  analysis  we  have  the  following 

RULE.  —  I.  When  the  fractions  have  the  same  denominator, 
subtract  the  less  numerator  from  the  greater,  and  place  the 
difference  over  the  common  denominator. 

II.  When  they  have  not  the  same  denominator,  reduce  them 
lo  a  common  denominator,  and  then  subtract  as  before. 

EXAMPLES. 
Make  the  following  subtractions  : 


1 .  From  -f-  take  f. 

2.  From  f  take  f. 

3.  From  -     take  - 


4.  From  1,  take  -fifo. 

5.  From  £  of  12,  take  ff  of  J. 

6.  F'mf  of  1J  of  7,  take  j.  off. 

7.  From  f  of  J  of  J  take  -ft  of  §  of  1. 

8.  From  £  of  J  of  6J,  take  f  of  f  of  f . 

9.  From  T*T  of  f£  of  J,  take  ^  of  ^. 

10.  What  is  the  difference  between  41  and 


OPERATION. 


or> 


16i  MULTIPLICATION  OF 

176.  Therefore  :  When  there  are  mixed  numbers,  change 
both  to  improper  fractions  and  subtract  as  in  Art.  11.5  ;  or, 
subtract  the  integral  and  fractional  numbers  separately,  and 
write  the  results. 

11.  From  S4-&  take  16J.       |    12.  From  246f  take  164£. 

13.  From  7£  take  4}  :  ^=1ft.  and  1=^. 

NOTE.  —  Since  we  cannot  take  &  from  -/,-  we  OPERATION. 

borrow  1,  or  ||,  from  the  minuend,  which  added  7*=7T&- 

to  ^r=H  J  then  ff  from  £f  leaves  f<f.     We  must  41  —4  V 
now  carry  1  to  the  next  figure  of  the  subtrahend 

and  proceed  as  in  subtraction  of  simple  numbers.  Ans.  2|-^ 

14.  From  16*  take  5f  16.  From  36f  take  27^. 

15.  From  26f  take  19f         It.  From  400T\  take  327*. 

18.  From  J  take  ^. 

NOTE.  —  When  the  numerators  are  1,  OPERATION. 

the  difference  of  the  two    fractions  is  l_Ti_=l-_.— 

equal  to  the  difference  of  the  denomina-  i  __  i    _ 
tors  divided  by  their  product 

19.  What  is  the  difference  between  ^  and  J  ?     Between 

iandTV?  ^and^V?  A  and  - 


MULTIPLICATION  OF  FRACTIONS. 

177.  MULTIPLICATION  of  Fractions  is  the  operation  of  taking 
one  number  as  many  times  as  there  are  units  in  another, 
when  one  of  the  numbers  is  fractional,  or  when  they  are  both 
fractional. 

1.  If  one  yard  of  cloth  cost  £  of  a  dollar,  what  will  4  yards 
cost? 

ANALYSIS.  —  Four   yards  will  cost  4  OPERATION. 

times  as  much  as  1  yard;    if  1  yard     J  x4r=A£±—  ^—  2J 
costs  5  eighths  of  a  dollar,  4  yards  will 

cost  4  times  5  eighths  of  a  dollar,  which  are  20  eighths  ;  equal  to 
2i  dollars. 

176.  When  there  are  mixed  numbers,  how  do  you  subtract?    Explain 
the  case  when  the  fractional  part  of  the  subtrahend  is  the  greater  ? 

177.  What  is  multiplication  of  fractions  ? 


COMMON   FKACTIOHS.  1G5 


OPERATION. 

W    /\    ^  tj    •     j    - 


2d.  If  we  divide  the  denominator  by  4,  OR, 

the  fraction  will  be  multiplied  by  4  (Prop.  o  • 


II) :  performing  the  operation,  we  obtain, 
which  —  2i :  hence, 


To  multiply  a  fraction  by  a  whole  number  :  —  Multiply  the 
numerator,  or  divide  the  denominator  by  the  multiplier. 


EXAMPLES. 


1.  Multiply  -^  by  12. 

2.  Multiply  |£  by  7. 

3.  Multiply  iff.  by  9. 


4.  Multiply  1T~JL  by  5. 

5.  Multiply  -J-ff  by  49. 

6.  Multiply  i^f  by  26. 

7.  If  1  dollar  will  buy  f  of  a  cord  of  wood,  how  much  will 
15  dollars  buy  ? 

8.  At  |  of  a  dollar  a  pound,  what  will  12  pounds  of  tea 
cost  ? 

9.  If  a  horse  cats  J  of  a  bushel  of  oats  in  a  day,  how  much 
will  18  horses  eat  ? 

10.  What  will  64  pounds  of  cheese  cost,  at  -^  of  a  dollar 
a  pound  ? 

11.  If  a  man  travel  2  of  a  mile  an  hour,  how  far  will  he 
travel  in  16  hours? 

12.  At  f  of  a  cent  a  pound,  what  will  45  pounds  of  chalk 
cost? 

13.  If  a  man  receive  -^  of  a  dollar  for  1  day's  labor,  how 
much  will  he  receive  for  15  days  ? 

14.  If  a  family  consume  ^  of  a  barrel  of  flour  in  1  month, 
how  much  will  they  consume  in  9  months  ? 

15.  If  a  person  pays  -j-J-  of  a  dollar  a  month  for  tobacco, 
how  much  does  he  pay  in  1 8  months  ? 

181.   To  multiply  a  whole  number  by  a  fraction. 
1.  At  15  dollars  a  ton,  what  will  |-  of  a  ton  of  hay  cost? 

ANALYSIS.— 1st.  Four  fifths  of  a  ton  will 
cost  4  times  as  much  as  1  fifth  of  a  ton ;  if  OPERATION. 

1  ton  cost  15  dollars,  1  fifth  will  cost  i  of  15     (15-i-5)x4  =  12 
dollars,  or  3  dollars,  and  i  will  cost  4  times  3     v 
dollars,  which  are  12  dollars. 


180.  How  do  you  multiply  a  fraction  by  a  whole  number  ? 


166  MULTIPLICATION  OF 

OR  :  2d.  4  fifths  of  a  ton  will  cost  1  fifth 

of  4  times  the  cost  of  1  ton  ;  4  times  15  is  60,       1  £  v  l  •  Z 10 

and  1  fifth  of  60  is  12. 


4 


NOTE. — Both  operations  may  be  combined  -"*  2 

in  one  by  the  use  of  the  vertical  line  and  can- 
cellation :  hence, 

|  12   Ans. 

Divide  the  whole  number  by  the  denominator  of  the  fraction 
and  multiply  the  quotient  by  the  numerator ; 

Or  :  Multiply  the  whole  number  by  the  numerator  of  the 
fraction  and  divide  the  product  by  the  denominator. 


EXAMPLES. 


1.  Multiply  24  by  ?. 

2.  Multiply  42  by 


3.  Multiply  105  by 

4.  Multiply  64  by 


5.  What  is  the  cost  of  •§•  of  a  yard  of  cloth  at  8  dollars  a 
yard  ? 

6.  If  an  acre  of  land  is  valued  at  75  dollars,  what  is  -^  of 
it  worth  ? 

7.  If  a  house  is  worth  320  dollars,  what  is  T9^-  of  it  worth  ? 

8.  If  a  man  travel   46  miles  in  a  day,  how  far  does  he 
travel  in  £  of  a  day  ? 

9.  At  18  dollars  a  ton,  what  is  the  cost  of  ^  of  a  ton  of 
hay? 

10.  If  a  man  earn  480  dollars  in  a  year,  how  much  does 
he  earn  in  -J-J  of  a  year  ? 

182.  To  multiply  one  fraction  by  another. 

1 .  If  a  bushel  of  corn  cost  f  of  a  dollar,  what  will  -f  of  a 
bushel  cost  ? 

OPERATION. 

ANALYSIS. — 5-sixths   of  a   bushel  will   cost        Jx|$.— ^.£—  «_ 
£   times    as    much    as   1   bushel,   or  5   times  4  ! 

1  sixth  of  a  bushel :    i  of  £  is  &,  (Art.   180),  g 

and  5  times  -fa  is  £$=$  :  hence, 


8     5  = 


181.  How  do  you  multiply  a  whole  number  by  a  fraction  ? 


COMMON   FRACTIONS.  167 

Multiply  the  numerators  together  for  a  new  numerator  and 
the  denominators  together  for  a  new  denominator. 

NOTES.— 1.  When  the  multiplier  is  less  than  1,  we  do  not  take 
the  whole  of  the  multiplicand,  but  only  such  a  part  of  it  as  the 
multiplier  is  of  1. 

2.  When  the  multiplier  is  a  proper  fraction,  multiplication  does 
not  imply  increase,  as  in  the  multiplication  of  Avhole  numbers. 
The  product  is  the  same  part  of  the  multiplicand  which  the  multi- 
plier is  of  1. 


EXAMPLES. 


1.  Multiply  I  by 

2.  Multiply  A  by 


3.  Find  the  pro't  of 

4.  Find  the  pro't  of  f  ft,  f  \. 


|,  J, 


5.  If  silk  is  worth  ft  of  a  dollar  a  yard,  what  is  f  of  a  yard 
worth  ? 

6.  If  I  own  ^  of  a  farm  and  sell  |  of  my  share,  what  part 
of  the  whole  farm  do  I  sell  ? 

7.  At  ±  of  a  dollar  a  pound,  what  will  ft  of  a  pound  of 
tea  cost  ? 

8.  If  a  knife  cost  *  of  a  dollar  and  a  slate  -f  as  much,  what 
does  the  slate  cost  ? 

OPERATION. 

9.  Multiply  5 J  by  -J-  of  |.        5^=^  ;  i  Of  f =- 

21   v     8    — -  7 

NOTE. — Before  multiplying,        *      3ir     I  ; 
reduce  both  fractions  to  the  form  *   i 

of  simple  fractions. 


9  |  1=1  Ans. 


GENERAL    EXAMPLES. 


1 .  Mult,  l  of  I  of  4-  by 

2.  Mult  i  by  $  of  If. 

3.  Mult.  J  of  3  by  i  of 


4.  Mult.  5  of  |  of  f  by  4J. 

5.  Mult.  14  of'-f  of  9  by  Gf 

6.  Mult,  f  of  6  of  -|  by  f  of  4. 

183.    When  the  multiplicand  is  a  whole  and  the  multi- 
plier  a  mixed  number. 


183.  How  do  you  multiply  one  fraction  by  another?  When  the 
multiplier  is  less  than  1,  what  part  of  the  multiplicand  is  taken  ?  If  the 
fraction  is  proper,  does  multiplication  imply  increase  ?  What  part  is  the 
product  of  the  multiplicand  ? 


168  DIVISION  OF 

7.  What  is  the  product  of  48  by  8£  ? 

NOTE.— First  multiply  48  by  •£,  which  gives  48  x  ±=  8 
8  ;  then  by  8,  which  gives  384,  and  the  sum,  392  40  v  Q  —  OQJ. 
is  the  product :  hence, 

392 

Multiply  first  by  the  fraction,  and  then  by  the  whole 
number,  and  add  the  products. 


8.  Mult.  67  by  9, 

9.  Mult.  12§  by 


10.  Mult.  108  by  1 

11.  Mult.  5f  by  3|. 


12.  What  is  the  product  of  6|,  2£  and  J  of  12  ? 

13.  What  will  24  yards  of  cloth  cost  at  3|  dollars  a  yard  ? 

14.  What  will  6 §  bushels  of  wheat  cost  at  3j  dollars  a 
bushel  ? 

15.  A  horse  eats  ^\  of  -£  of  12  tons  of  hay  in  three  months  ; 
how  much  did  he  consume  ? 

16.  Jf  §  of  £  of  a  dollar  buy  a  bushel  of  corn,  what  will 
^  of  T6T  of  a  bushel  cost  ? 

17.  What  is  the  cost  of  5|  gallons  of  molasses  at  96 J  cents 
a  gallon  ? 

18.  What  will  7|£  dozen  caudles  cost  at  T3T  of  a  dollar  per 
dozen  ? 

19.  What  must  be  paid  for  175  barrels  of  flour  at  7|  dol- 
lars a  barrel  ? 

20.  If  |  of  -f-  of  2  yards  of  cloth  can  be  bought  for  one  dol- 
lar, how  much  can  be  bought  for  |  of  13|  dollars  ? 

21.  What  is  the  cost  of  15|  cords  of  wooc^at  3|-  dollars  a 
cord? 

DIVISION  OF  FRACTIONS. 

184.  Division  of  Fractions  is  the  operation  of  finding  a 
number  which  multiplied  by  the  divisor  will  produce  the  divi- 
dend, when  one  or  both  of  the  parts  are  fractional. 

185.  To  divide  a  fraction  by  a  ivhole  number. 

1.  If  4  bushels  of  apples  cost  -jj-  of  a  dollar,  what  will 
1  bushel  cost  ? 

183.  How  may  you,  multiply  when  the  multiplicand  is  a  icJiolc  and  the 
multiplier  a  mixed  number? 

184.  What  is  division  of  fractions? 

185.  How  do  you  divide  a  fraction  by  a  whole  number  ? 


COMMON   FRACTIONS. 


169 


ANALYSIS. — Since  4  bushels  cost  f  of  a  dollar, 
J.  bushel  will  cost  \  of  f  of  a  dollar.  Dividing 
the  numerator  of  the  fraction  f  by  4,  we  have 
§  (Art.  159). 


OPERATION. 


Multiplying  the  denominator  by  4  will   pro-  A -^-4 — -^j~~§- 
duce  the  same  result  (Art.  160) :  hence, 

Divide  the  numerator  or  multiply  the  denominator  by  the 
divisor. 


NOTE. — By  the  use  of  the  vertical  line  and  the 
principles  of  cancellation  (Art.  148),  all  operations 
in  divisions  of  fractions  may  be  greatly  abridged. 




9  |  2=f 


EXAM 

1.  Divide  ff-  by  6. 
3.  Divide  ^f-  by  9. 
3.  Divide  ^  by  15. 
4.  Divide  -fff  by  75. 

PLES. 

5.  Divide  ||  by  6. 
6.  Divide  ££  by  12. 
7.  Divide  if  by  20. 
8.  Divide  iff  by  27. 

9.  If  6  horses  eat  T^j  of  a  ton  of  hay  in  1  month,  how  much 
will  one  horse  eat  ? 

10.  If  9  yards  of  ribbon  cost  f  of  a  dollar,  what  will  1  yard 
cost? 

11.  If  1  yard  of  cloth  cost  4  dollars,  how  much  can  be 
bought  for  f  of  a  dollar  ? 

12.  If  5  pounds  of  coffee  cost  if  of  a  dollar,  what  will 
1  pound  cost  ? 

13.  At  $6  a  barrel,  what  part  of  a  barrel  of  flour  can  be 
bought  for  -f  of  a  dollar  ? 

14.  If  10  bushels   of  barley  cost   3J   dollars,  what  will 
1  bushel  cost  ? 


NOTE. — We  reduce  the  mixed  number  to 
an  improper  fraction  and  divide  as  in  the 
case  of  a  simple  fraction. 


OPERATION. 


J/-f-10  =  i  Ans. 

15.  If  21  pounds  of  raisms  cost  4|  dollars,  what  will  1 
pound  cost  ? 

16.  If  12  men  consume  6f  pounds  of  meat  in  a  day;  how 
much  does  1  man  consume  ? 


170  DIVISION    OF 

186.  To  divide  a  whole  number  by  a  fraction. 

I.  At  f  of  a  dollar  apiece,  how  many  hats  can  be  bought 
for  6  dollars  ? 

ANALYSIS.— Since  £  of  a  dollar  will  OPERATION. 

buy  one  hat,  6  dollars  will  buy  as  many       6-=-4-= 6x5-f-4  =  7i. 
hats  as  £  is  contained  times  in  6  ;  and 
as  there  are  5  times  as  many  fifths  as 
whole  things  in  any  number,  in  6  there 
are  30  fifths,  and  4  fifths  is  contained  in  2  ; 

30  fifths  7i  times :  hence,  _ 

Invert  the  terms  of  the  divisor  and  multiply  the  whole  num- 
ber by  the  new  fraction. 

EXAMPLES. 

1.  Divide  14  by  J.  3.  Divide  63  by  £f . 


2.  Divide  212  by 


4.  Divide  420  by 


5.  At  -^  of  a  dollar  a  yard,  how  many  yards  of  cloth  can 
be  bought  for  9  dollars  ? 

6.  If  a  man  travel  ^  of  a  mile  in  1  hour,  how  long  will  it 
take  him  to  travel  10  miles  ? 

7.  If  y  of  a  ton  of  hay  is  worth  9  dollars,  what  is  a  ton 
worth  ? 

187.  To  divide  one  fraction  by  another. 

1.  At  f  of  a  dollar  a  gallon,  how  much  molasses  can  be 
bought  for  |  of  a  dollar  ? 

ANALYSIS. — Since  §   of  a   dollar  OPERATION. 

will  buy  1  gallon,  I  of  a  dollar  will        J-T-?-=  I  x  4^^4 

--  -  ---  "  ^     " 


buy  as  many  gallons  as  \  is  contained  g 

times  in  \  :  one  is  contained  in  I,  I  o 

times :  but  •&•  is  contained  5  times  as 

many  times  as  1,  or  *•£-  times  ;  but  2  161 

fifths  is  contained  half  as  many  times 

as  i,  or  f  $  times,  equal  to  2-13-j  times  :  hence, 


I.  Invert  the  terms  of  the  divisor. 

II.  Multiply  the  numerators  together  for  the  numerator 
of  the  quotient,  and  the  denominators  together  for  the  de- 
nominator of  the  quotient. 

186.  How  do  you  divide  a  whole  number  by  a  fraction  ? 


COMMON   FRACTIONS.  171 

NOTES. — 1.  If  the  vertical  line  is  used,  the  denominator  of  the 
dividend  and  the  numerator  of  the  divisor  fall  011  the  left,  and  the 
other  terms  on  the  right. 

2.  Cancel  all  common  factors. 

3.  If  the   dividend  and  divisor  have  a  common   denominator, 
they  will  cancel,  and  the  quotient  of  their  numerators  will  be  the 
answer. 

4.  When  the  dividend  or   divisor   contains   a  whole  or  mixed 
number,  or  compound  fractions,  reduce  them  to  tiie  form  of  simple 
fractions  before  dividing. 


EXAMPLES. 


1.  Divide  -ft  by  ft. 

2.  Divide  -ft  by  Tf . 

3.  Divide  3£  by  |f 


4.  Divide  }  of  f  by  T^  of  1J. 

5.  Divide  f  of  21  by  f  of  3|. 

6.  Divide  6|  by  2J. 


7.  At  l  of  a  dollar  a  pound,  how  much  butter  can  be 
bought  for  |£  of  a  dollar  ? 

8.  If  1  man  consume  1^  pounds  of  meat  in  a  day,  how 
many  men  would  8J-  pounds  supply  ? 

9.  If  6  pounds  of  tea  cost  4J  dollars,  what  does  it  cost  a 
pound  ? 

10.  At  it  of  a  dollar  a  basket,  how  many  baskets  of  peaches 
can  be  bought  for  11^  dollars  ? 

11.  If  £  of  a  ton  of  coal  cost  6|  dollars,  what  will  1  ton 
cost,  at  the  same  rate  ? 

12.  How  much  cheese  can  be  bought  for  -J£  of  a  dollar  at 
£  of  a  dollar  a  pound  ? 

13.  A  man  divided  2f  dollars  among  his  children,  giving 
them  y7^  of  a  dollar  a  piece  ;  how  many  children  had  he  ? 

14.  How  many  times  will  •J-J-  of  a  gallon  of  beer  fill  a  vessel 
holding  i  of  f  gallons  ? 

15.  How  many  tunes  is  £  of  ^  of  27  contained  in  -£  of  i 
of42§? 

16.  If  5-J-  bushels  of  potatoes  cost  2f  dollars,  how  much  do 
they  cost  a  bushel  ? 

17.  If  John  can  walk  21  miles  in  -^  of  a  day,  how  far  can 
he  walk  in  1  day  ? 

18.  If  a  turkey  cost  If  dollars,  how  many  can  be  bought 
for  12f  dollars  ? 

19.  At  f  of  |  of  a  dollar  a  yard,  how  many  yards  of  rib- 
bon can  be  bought  for  -|i  of  a  dollar  ? 

187.  How  do  you  divide  one  fraction  by  another  9 


172  REDUCTION   OF 

REDUCTION  OP  COMPLEX  FRACTIONS. 

188.  Complex  Fractions  are  only  other  forms  of  expression 

for  the  division  of  fractions  :  thus  ;  1  is  the  same  as  %  divided 

by  -?j  ;  and  may  be  written,  %  x  f =f£ =2^-. 

181).  To  reduce  a  complex  fraction  to  the  form  of  a  sim- 
ple fraction. 

1.  Reduce  _£  to  its  simplest  form. 

*i 

OPERATION. 
4 

?j^-¥-!=^xA=Ts^  Ans.-,  hence, 

42  —  TJ-  09 

3 

RULE. — Divide  the  numerator  of  the  complex  fraction  by  its 
denominator, 

Or  :  Multiply  the  numerator  of  the  upper  fraction  into  the 
denominator  of  the  loiuer,for  a  numerator  ;  and  the  denomi- 
nator of  the  upper  fraction  into*  the  numerator  of  the  lower,  for 
a  denominator.  9 

NOTES. — 1.  When  either  of  the  terms  of  a  complex  fraction  is  a 
mixed  number,  or  compound  fraction,  it  must  first  be  reduced  to 
the  form  of  a  simple  fraction. 

2.  When  the  vertical  line  is  used,  the  numerator  of  the  upper  and 
the  denominator  of  the  lower  numbers  fall  on  the  right  of  the  verti- 
cal line,  and  the  other  terms  on  the  left. 

EXAMPLES. 

Reduce  the  following  complex  fractions  to  their  simplest  form : 


1.  Reduce  jL 

2.  Reduce  ^1 


3.  Reduce 


4.  Reduce  f  of  i. 


5   Reduce 


6.  Reduce  f£. 
8f 

1.  Reduce 


8.  Reduce 


__ 

*  of  15 

214f 

25H 


9.  Reduce  '^—, 


10.  Reduce 


of  48 


DENOMINATE    FRACTIONS. 


173 


DENOMINATE   FRACTIONS. 

190.  A  DENOMINATE  Fraction  is  one  in  which  the  unit  of 
the  fraction  is  a  denominate  number.     Thus,  f  of  a  yard  is  a 
denominate  fraction. 

191.  REDUCTION  of  denominate  fractions  is  the  operation 
of  changing  a  fraction  from  one  denominate  unit  to  another 
without  altering  its  value. 

There  are  four  cases  : 

1st.  To  change  from  a  greater  unit  to  a  less,  as  from  yards 
to  inches  : 

2d.  To  change  from  a  less  unit  to  a  greater  : 

3d.  To  find  the  value  of  a  fraction  in  integers  of  lower 
denominations  : 

4th.  To  find  the  value  of  integers  in  a  fraction  of  a  larger 
unit. 

These  cases  will  be  arranged  in  sets  of  two  and  two. 


192.  To  change  from  a 
greater  unit  to  &  less. 

1.  In  $  of  a  yard,  how 
many  inches  ? 

OPERATION. 

f  x  3  x  12=if£=20  inches. 

ANALYSIS.— Since  in  1  yard 
there  are  3  feet,  in  f  yards  there 
are  $  times  3  feet=-^-  feet.  And 
since  in  1  foot  there  are  12 
inches,  in  ^  feet  there  are  19- 
times  12  inches =I0a= 20  inch's  : 
hence, 

RULE. — Multiply  the  frac- 
tion and  the  products  which 
arise  by  the  units  of  the  scale, 
in  succession,  until  you  reach 
the  unit  required. 


193.  To  change  from  a 
less  unit  to  a  greater. 

1.  In  20  inches,  how  many 
yards  ? 

OPERATION. 

20  xAxi=H=*  Jards< 
ANALYSIS. — Since  12  inches 
make  1  foot,  in  20  inches  there 
are  as  many  feet  as  12  inches  is 
contained  times  in  20  inches 
=  H  feet;  and  as  3  feet  make 
1  yard,  in  ^  feet  there  are  as 
many  yards  as  3  feet  is  contained 
times  in  ^  fect=|§=f  yards: 
hence, 

RULE. — Divide  the  fraction 
and  the  quotients  which  arise, 
by  the  units  of  the  scale,  in  suc- 
cession, until  you  reach  the 
unit  required. 


188.  What  are  complex  fractions? 

189.  How  do  you  reduce  complex  to  simple  fractions  ? 


174  DENOMINATE   FRACTIONS. 

NOTE. — In  every  operation  of  reduction,  in  which  there  are 
common  factors,  be  sure  and  cancel  them  before  making  the  final 
multiplication. 

EXAMPLES. 

1.  Reduce  -g-f-g-  of  a  hogshead  to  the  fraction  of  a  quart. 

2.  Reduce  -^  of  a  bushel  to  the  fraction  of  a  pint. 

3.  Reduce  -g^ir  of  a  pound  Troy  to  the  fraction  of  a  grain. 

4.  What  part  of  a  foot  is  -J-&TF  of  a  furlong  ? 

5.  What  part  of  a  minute  is  -^Vo-  °f  a  day  ? 

6.  Reduce  ^Vjizr  °f  a  cwt.  to  the  fraction  of  an  ounce. 

7.  Reduce  f  of  a  gallon  to  the  fraction  of  a  hogshead. 

8.  What  part  of  a  £  is  £  of  a  shilling  ? 

9.  What  part  of  a  hogshead  is  -g-  of  a  quart  ? 

10.  What  part  of  a  mile  is  -fr  of  a  foot  ? 

11.  Reduce  4-^0  of  £  to  the  fraction  of  a  farthing. 

12.  Reduce  yV  of  an  Ell  Eng.  to  the  fraction  of  a  nail. 

13.  Reduce  |-  of  a  nail  to  the  fraction  of  a  yard  ? 

14.  Reduce  J  of  %  of  a  foot  to  the  fraction  of  a  mile. 

15.  Reduce  5^7  6  of  a  ton  to  the  fraction  of  a  pound. 

16.  Reduce  J[  of  3|  pwt.  to  the  fraction  of  a  pound  Troy. 
It.  What  part  of  a  mile  is  j  of  a  rod  ? 

18.  What  part  of  an  ounce  is  -fo  of  a  scruple  ? 

19.  -^f-g-  of  a  day  is  what  portion  of  10  minutes? 

20.  What  part  of  J-  of  a  foot  is  yf-g-  of  a  furlong  ? 

21.  Reduce  -g^g-  of  a  hogshead  of  ale  to  the  fraction  of  a 
pint. 

190.  What  is  a  denominate  fraction  ? 

191.  What  is  reduction  of  denominate  fractions?    How  many  casca 
are  there  V    Name  them. 

192.  How  do  you  change  from  a  greater  unit  to  a  less  ? 

193.  How  do  you  change  from  a  less  unit  to  a  greater  ? 


DENOMINATE    FRACTIONS. 


175 


194.  To  find  the  value  of 
a  fraction  in  integers  of  loiver 
denominations. 

1.  What  is  the  value  of  f 
of  a  pound  Troy  ? 

ANALYSIS.— £  of  a  pound  re- 
duced to  the  fraction  of  an  ounce 
is  |xl2=^».  of  an  ounce,  (Art. 
177.),  which  is  equal  to  9§- 
ounces  :  f  of  an  ounce  reduced 
to  the  fraction  of  a  pennyweight 
is  |  x  20=^  of  a  pwt.,  or  12pwt. 

OPERATION. 

burner.     4 

12  oz.     pwt. 

Denom.       5)48(9... 12 
45 
3 
20 

5)60 
60 

RULE.  —  I.  Multiply  the 
numerator  of  the  fraction  by 
the  number  which  will  re- 
duce it  to  the  next  lower  de- 
nomination and  divide  the 
product  by  the  denominator. 

II.  If  there  is  a  remain- 
der, reduce  it  in  the  same 
manner,  and  so  on,  till 
the  lowest  denomination  is 
obtained. 


195.  To  find  ike  value  of 
integers  in  a  fraction  of  a 
higher  denomination. 

2.  Reduce  9oz.  12pwts.  to 
the  fraction  of  a  pound  Troy. 

ANALYSIS. — In  1  pound  there 
are  240  pennyweights:  1  pen- 
ny  weight  is  ^  of  a  pound  ;  and 
9  ounces  12pwts.  =  l&Zpwts.  is 
of  a  pound=£  of  a  pound. 


OPERATION. 

1  lb.     oz.  pwts. 
12        9..  12 
12        20 
20  Num-  l92_ 
240  Denom.  §40" ~ 


RULE. — I.  Reduce  the  given 
ntegers    to    the    lowest    de- 
nomination  named,  and  the 
result  will  be  the  numerator 
jf  the  required  fraction. 

II.  Eeduce  1  unit  of  the 
required  denomination,  to  the 
denomination  of  the  numera- 
or,  and  the  result  will  be 
he  denominator  of  the  re- 
quired fraction. 


EXAMPLES. 

3.  What  is  the  value  of  -£  of  a  tun  of  wine  ? 

4.  What  part  of  a  tun  of  wine  is  3hhd.  Slgal.  2gt.  ? 

194.  How  do  you  find  the  value  of  a  fraction  in  integers  of  lower  de- 
nominations ? 

195.  How  do  yon  find  the  value  of  integers  in  a  fraction  of  a  higher 
denomination  ? 


176  ADDITION   AND   SUBTRACTION    OF 


5.  What  is  the  value  of  y9^  of  a  yard  ? 

6.  What  is  the  value  of  -|  of  a  month  ? 

7.  What  is  the  value  of  f  of  a  chaldron  ? 

8.  What  is  the  value  of  %  of  a  mile  ? 

9.  What  is  the  value  of  -fe  of  a  ton  ? 

10.  What  is  the  value  of  $  of  3  days  ? 

11.  What  is  the  value  of  £  of  £  of  6  §  bushels  of  grain  ? 

12.  Reduce  Sgals.  2qts.  to  the  fraction  of  a  hogshead. 

13.  Reduce  2fur.  36rd  2yd.  to  the  fraction  of  a  mile. 

14.  What  part  of  a  £  is  5s.  *I±d.  ? 

15.  What  part  of  a  pound  Troy  is  lOoz.  13pwt.  Sgr.  ? 

16.  llcwt.  Qqr.  12/6.  7  02.  l%dr.  is  what  part  of  a  ton? 

17.  What  part  is  2pk.  ±qt.  of  Ibu.  Spk.  ? 

18.  24/6.  6oz.  is  what  part  of  Zqr.  12/6.  I2oz.  ? 

19.  Reduce  3wk.  Id.  9/i.  36?n.  to  the  fraction  of  a  month 

20.  Reduce  2E.  32rrf.  8z/<7.  to  the  fraction  of  an  acre. 

21.  Reduce  12s.  $d.  \\far.  to  the  fraction  of  a  guinea. 

22.  What  is  the  value  of  Ty&,  apothecaries'  weight  ? 

23.  What  part  of  an  Ell  English  is  3qr.  3?ia.  l\in.  ? 

24.  What  is  the  value  of  $hhd.  ? 

25.  What  is  the  value  gf  f  of  3  barrels  of  beer  ? 

26.  What  is  the  value  of  TV  of  a  cwt.  ? 

27.  Reduce  3°  15'  18|"  to  the  fraction  of  a  sign. 

28.  Reduce  3£  inches  to  the  fraction  of  a  hand. 

29.  What  is  the  value  of  -fa  of  a  hogshead  of  wine  ? 

30.  What  is  the  value  of  7    of  an  acre  of  land  ? 


ADDITION  AND  SUBTRACTION. 
196.  To  add  or  subtract  denominate  fractions. 
1.  Add  §  of  a  £  to  £  of  a  shilling. 

|  of  a  £=§  of  2^=*$-  of  a  shilling. 
Then,  4J*  +  f  ^W+lf  =W*=  ¥*•  =  14s  2^ 


196.  Give  the  rule  for  adding  and  subtracting  denominate  fractions. 


DENOMINATE   FRACTIONS.  177 

Or,  the  |-  of  a  shilling  may  be  reduced  to  the  fraction  of  a  £> : 
thus, 

I  °f  ^V=Tth>  of  a  &=•&  of  a  £  : 
then,  S+A  =  H+A=H  of  a  £> 

which  being  reduced,  gives  14s.  %d.  Ans. 

2.  Add  f  of  a  year,  |  of  a  week,  and  |  of  a  day. 

•f  of  a  year=f  of -^p  days=31w&.  2da. 
J  of  a  week=J  of  7  days     —  -    -    2da.  Shr. 
I  of  a  day  =    -    -    -    -     =  -    -         -  3/tr. 
Ans.  Slwk.  Ida.  llhr. 

3.  From  \  of  a  £  take  J  of  a  shilling. 

J  of  a  shilling^  of  -5^  of  a  £=-fa  of  a  £. 
Then,    '         i— AF=^-A=-»ofa^=9»-  8^ 

4.  From  1  j#>.  Troy  weigfit,  take  ^oz. 

Ib.  oz.  pwt.  gr. 

lJ/6.=£  of  Jjao2=21oz.  =  l  9 

•Joz.=^  of-y-  ofygrr.  =  80gfr.  =  0  038 

J?is.  1  8     16     16 

RULE. — Reduce  the  given  fractions  to  the  same  unit,  and 
then  add  or  subtract  as  in  simple  fractions,  after  ivhich  reduce 
to  integers  of  a  lower  denomination  : 

Or  :  Reduce  the  fractions  separately  to  integers  of  lower  de- 
nominations, and  then  add  or  subtract  as  in  denominate  num" 
bers. 

EXAMPLES. 

5.  Add  1J  miles,  T^  furlongs,  and  30  rods. 

6.  Add  §  of  a  yard,  J  of  a  foot,  and  $  of  a  mile. 

7.  Add  |  of  a  cwt.,  *£  of  a  Ib.,  13oz.,  J  of  a  curt,  and  6/6. 

8.  From  J  of  a  day  take  f  of  a  second. 

9.  From  |  of  a  rod  take  f  of  an  inch. 

10.  From  *fc  of  a  hogshead  take  f  of  a  quart. 

11.  From  $oz.  take  %pwl. 

12.  From  4fcw£.  take  4Ty&. 

12 


178  DUODECIMALS. 

13.  Mr.  Merchant  bought  of  farmer  Jones  22J  bushels  of 
wheat  at  one  time,  19^  bushels  at  another,  and  33f  at  an- 
other :  how  much  did  he  buy  in  all  ? 

14.  Add  %  of  a  ton  and  -fa  of  a  cwt. 

15.  Mr.  Warren  pursued  a  bear  for  three  successive  days  ; 
the  first  day  he  travelled  28-f-  miles  ;  the  second  33T^  miles  ; 
the  third  29-^j-  miles,  when  he  overtook  him  :  how  far  had  he 
travelled  ? 

16.  Add  5f  days  and  52  T%-  minutes. 

17.  Add  $cwt.,  S%lb.,  and  3Ty&. 

18.  A  tailor  bought  3  pieces  of  cloth,  containing  respect- 
ively, 18|  yards,  21|  Ells  Flemish,  and  16f  Ells  English  : 
how  many  yards  in  all  ? 

19.  Bought  3  kinds  of  cloth  ;  the  first  contained  \  of  3  of 
f  of  £  yards  ;  the  second,  £  of  f  of  5  yards  ;  and  the  third,  \ 
of  f  of  |  yards  :  how  much  in  them  all  ? 

20.  Add  \\cwt.  17f/&.  and  7foz. 

21.  From  f  of  an  oz.  take  £  of  &pwt. 

22.  Take  }  of  a  day  and  J  of  §  of  j  of  an  hour  from 
3 1  weeks. 

23.  A  man  is  6|  miles  from  home,  and  travels  4wi.  Ifur. 
24?*d.,  when  he  is  overtaken  by  a  storm  :  how  far  is  he  then 
from  home  ? 

24.  A  man  sold  -J^  of  his  farm  at  one  time,  ^£  at  another, 
and  ^7  at  another  :  what  part  had  he  left  ? 

25.  From  1 J  of  a  £  take  |  of  a  shilling. 

26.  From  l£oz.  take  %pwt. 

27.  From  8%cwt.  take  4Ty6. 

28.  From  3|Z6.  Troy  weight,  take  \pz. 

29.  From  1^  rods  take  ^  of  an  inch. 

30.  From  $f  g)  take  ^  3  . 

DUODECIMALS. 

197.  If  the  unit  1  foot  be  divided  into  12  equal  parts,  each 
part  is  called  an  inch  or  prime,  and  marked  '.  If  an  inch  be 
divided  into  12  equal  parts,  each  part  is  called  a  second,  and 
marked  ".  If  a  second  be  divided,  in  like  manner,  into  12 


DUODECIMALS.  179 

equal  parts,  each  part  is  called  a  third,  and  marked  "'  ;  and 
so  on  for  divisions  still  smaller. 

This  division  of  the  foot  gives 

1'     inch  or  prime  -    -  ,  -    -  -  =  -^      of  a  foot. 

I"    second  is  ^  of  •&   -    -  -  =  y^    of  a  foot. 

1'"  third  is  TV  of  •&  of  A'  -  =  TT^  of  a  foot- 


NOTE.  —  The  marks  ',  ",  '",  &c.,  which  denote  the  fractional 
units,  are  called  indices, 

TABLE. 

12'"  make  1"  second. 

12"  "  1'  inch  or  prime. 

12'  "  1  foot. 

Hence  :  Duodecimals  are  denominate  fractions,  in  which 
the  primary  unit  is  1  foot,  and  12  the  scale  of  division. 

NOTE.  —  Duodecimals  are  chiefly  used  in  measuring  surfaces  and 
solids. 

ADDITION  AND  SUBTRACTION. 

198.  The  units  of  duodecimals  are  reduced,  added,  and 
subtracted,  like  those  of  other  denominate  numbers.  The 
scale  is  always  12. 

EXAMPLES. 

1.  In  185',  how  many  feet  ? 

2.  In  250",  how  many  feet  and  inches  ? 

3.  In  4367'",  how  many  feet? 

4.  What  is  the  sum  of  3/35.  6'  3"  2'"  and  2ft.  I'  10"  11'"? 

5.  What  is  the  sum  of  8/3L  9'  7"  and  6/fc.  7'  3"  4"'  ? 

6.  What  is  the  difference  between  9/fc.  3'  5"  6'"  and  7/35. 
3'  6"  7'"? 

7.  What  is  the  difference  between  40/35.  6'  6"  and  29/fc.  7'"  ? 

8.  What  is  the  difference  between  12ft.  7'  9"  6'"  and  4/2. 
9'  7"  9'"? 

197.  If  1  foot  be  divided  into  twelve  equal  parts,  what  is  each  part 
called  ?    If  the  inch  be  so  divided,  what  is  each  part  called  ?    What  are 
duodecimals  ?    For  what  are  duodecimals  chiefly  used  ? 

198.  How  do  you  add  and  subtract  duodecimals  ?    What  is  the  scale  ? 


180  DUODECIMALS. 

MULTIPLICATION. 

199.  Begin  with  the  highest  unit  of  the  multiplier  and  the 
lowest  of  the  multiplicand,  and  recollect, 

1st.  That  1  foot  x  1  foot=l  square  foot  (Art.  110). 
2d.    That  a  part  of  a  foot  x  a  part  of  a  foot = some  part  of  a 
square  foot. 

NOTE. — Observe  that  the   unit  is  changed,  by  multiplication, 
from  a  linear  to  a  superficial  unit. 

Multiply  6ft.  T  8"  by  2/fc.  9'. 

OPERATION. 

ANALYSIS.— Since  a  prime  is  ^  of  a  ft. 

foot  and  a  second  T^T,  g   y   g" 

2  x  8"  =-iiA  of  a  square  foot ;  which  re-  9   Q/ 
duced    to   12ths,  is  1'  and  4" :  that  is, 


1  twelfth,  and  4  twelfths  of  -fe  of  a  2  X  8"=        1'  4" 

square  foot.  2x7'=    1    2' 

2x7'  =14  twelfths=l/£.  2'  2  X  6   =12 

2x6  =12  square  feet,  9'  x  g"  —  6" 

9  x  8"=T^|-8  of  a  square  foot=6"  9'  x  7'  —        5'  3" 

9'xT=fA-=5'  3"  9'X6   =  4   6' 
9x6'=f|=46'  prod      18  3'   r 

RULE.  —  I.  Write  the  multiplier  under  the  multiplicand, 
so  that  units  of  the  same  order  shall  fall  in  the  same 
column. 

II.  Begin  with  the  highest  unit  of  the  multiplier   and 
the  lowest  of  the  multiplicand,  and  make  the  index  of  each 
product  equal  to  the  sum  of  the  indices  of  the  factors. 

III.  Eeduce  each  product,  in  succession,  to  the  next  higher 
denomination,  when  possible. 

NOTE.  —  The  index  of  the  unit  of  any  product  is  equal  to  the 
of  the  indices  of  the  factors. 


EXAMPLES. 

1  .  How  many  solid  feet  in  a  stick  of  timber  which  is  25 
feet  6  inches  long,  2  feet  7  inches  broad,  and  3  feet  3  inches 
thick  ? 

199.  Explain  the  method  of  multiplying  duodecimals.  Give  the 
rule. 


DUODECIMALS.  181 

OPERATION. 
.# 

Beginning  with  the  2  feet,  we  say  2  25      6'  length, 
times  6'  are  12'=1    square  foot :  then,  2  27'  breadth. 

times  25  are  50,  and  1  to  carry  are  51 f 

square  feet.  51      0 

Next,  7  times  6'  are  42", =3'  and  6"  :  3'  6' 

then  7'  times  25=175'=14  7':  hence,  the  ^4      »j' 
surface  is  65  10'  6",  and  by  multiplying  • 

by  the  thickness,  we  find  the  solid  contents  65    1"    o 
to  be  214  1'  1"  6'"  cubic  feet.  3     X  thickness. 

197     7'  6" 

16     5'  7"  6"'' 
214     1'1"6'" 

2.  Multiply  9/2.  4m.  by  8/2.  3m. 

3.  Multiply  9#.  2m.  by  fyfc  6m. 

4.  Multiply  24/2.  10m.  by  6/2.  8m. 

5.  Multiply  70/2.  9m.  by  12/2.  3m. 

6.  How  many  cords  and  cord  feet  in  a  pile  of  wood  24  feet 
long,  4  feet  wide,  and  3  feet  6  inches  high  ? 

7.  How  many  square  feet  are  there  in  a  board  17  feet  6 
inches  in  length,  and  1  foot  7  inches  in  width  ? 

8.  What  number  of  cubic  feet  are  there  in  a  granite  pillar 
3  feet  9  inches  in  width,  2  feet  3  inches  in  thickness,  and  12 
feet  6  inches  in  length  ? 

9.  There  is  a  certain  pile  of  wood,  measuring  24  feet  in 
length,   16    feet  9  inches    high,   and    12    feet  6  inches  in 
width.     How  many  cords  are  there  in  the  pile  ? 

10.  How  many  square  yards  in  the  walls  of  a  room,  14 
feet  8  inches  long,  11  feet  6  inches  wide,  and  7  feet  11  inches 
high  ? 

11.  If  a  load  of  wood  be  8  feet  long,  3  feet  9  inches  wide, 
and  6  feet  6  inches  high,  how  much  does  it  contain  ? 

12.  How  many  cubic  yards  of  earth  were  dug  from  a  cellar 
which  measured  42  feet  10  inches  long,  12  feet  6  inches  wide, 
and  8  feet  deep  ? 

13.  What  will  it  cost  to  plaster  a  room  20  feet  6'  long,  15 
feet  wide,  9  feet  6'  high,  at  18  cents  per  square  yard? 

14.  How  many  feet  of  boards  1  inch  thick  can  be  cut  from 
a  plank  18/2.  9m.  long,  l/t.  Sin.  wide,  and  3m.  thick,  if  there 
is  no  waste  in  sawing  ? 


182  DECIMAL  FRACTIONS. 

DECIMAL    FRACTIONS. 

200.  There  are  two  kinds  of  Fractions  :   Common  Frar 
tions  and  Decimal  Fractions. 

A  Common  Fraction  is  one  in  which  the  unit  is  divided 
into  any  number  of  equal  parts. 

A  Decimal  fraction  is  one  in  which  the  unit  is  divided  ac- 
cording to  the  scale  of  tens. 

201.  If  the  unit  1  be  divided  into  10  equal  parts,  the  parts 
are  called  tenths. 

If  the  unit  1  be  divided  into  one  hundred  equal  parts,  the 
parts  are  called  hundredths. 

If  the  unit  1  be  divided  into  one  thousand  equal  parts,  the 
parts  are  called  thousandths,  and  we  have  similar  expressions 
for  the  parts,  when  the  unit  is  further  divided  according  to  the 
scale  of  tens.  * 

These  fractions  may  be  written  thus  : 

Four-tenths,  -----  *fo. 

Six-tenths,     -  -  TV 

Forty-five  hundredths, 

125  thousandths, 

1047  ten  thousandths,  - 

From  which  we  see,  that  in  each  case  the  denominator 
indicates  the  fractional  unit  ;  that  is,  determines  whether  it  is 
one-tenth,  one-hundredth,  one-thousandth,  &c. 

202.  The  denominators  of  decimal  fractions  are  seldom 
written.     The  fractions  are  usually  expressed  by  means  of 
a  period,  placed  at  the  left  of  the  numerator. 

Thus      ^5-  is  written    -  .  4 


200.  How  many  kinds  of  fractions   are   there?     What   are   they? 
What  is  a  common  fraction  ?    What  is  a  decimal  fraction  ? 

201.  When  the  unit  1  Is  divided  into  10  equal  parts,  what  is  each 
part  called  ?    What  is  each  part  called  when  it  is  divided  into  100  equal 
parts?    When  into  10000?    Into  10,000,  &c.  ?    How  are  decimal  frac- 
tions formed  ?    What  gives  denomination  to  the  fraction  ! 


DECIMAL    FRACTIONS.  183 

This  method  of  writing  decimal  fractions  is"  a  mere  lan- 
guage, and  is  used  to  avoid  writing  the  denominators.  The 
denominator,  however,  of  every  decimal  fraction  is  always 
understood : 

It  is  the  unit  1  with  as  many  ciphers  annexed  as  there 
are  places  of  figures  in  the  decimal. 

The  place  next  to  the  decimal  point,  is  called  the  place 
of  tenths,  and  its  unit  is  1  tenth.  The  next  place,  to  the 
right,  is  the  place  of  hundredths,  and  its  unit  is  1  hundreth  ; 
the  next  is  the  place  of  thousandths,  and  its  unit  is  1  thous- 
andth ;  and  similarly  for  places  still  to  the  right. 

DECIMAL  NUMERATION  TABLE. 


d 
•S 

T3 

«  2 

£  |        oJ 

'o  a  2  'O'fsS 
«'g  £,§  Sg^ 
rS  "3  a  -*•'  ^  2  » 

a  a 


.4  is  read  4  tenths, 

.54  -  -  54  hundredths. 

.064  -  -  64  thousandths. 

.6754  -  -  6154  ten  thousandths, 

.01234  -  -  1234  hundred  thousandths 

.007654  -  -  7654  mfflionths. 

.0043604  -  -  43604  ten  millionths. 

NOTE.  —  Decimal  fractions  are  numerated  from  left  to  right  ; 
thus,  tenths,  hundredths,  thousandths,  &c. 

202.  Are  the  denominators  of  decimal  fractions  generally  written  ? 
How  are  the  fractions  expressed?  Is  the  denominator  understood?. 
What  is  it  ?  What  is  the  place  next  the  decimal  point  called  ?  What 
is  its  unit  ?  What  is  the  next  place  called  ?  What  is  its  unit  ?  What 
is  the  third  place  called  ?  What  is  its  unit  ?  Which  way  are  decimals 
numerated  ? 


184  DECIMAL    FRACTIONS. 

203.  Wfite  and  numerate  the  following  decimals  : 

Four  tenths,  .4 

Four  hundredths,  -  .0  4 

Four  thousandths,  .004 

Four  ten  thousandths,      -  .0004 

Four  hundred  thousandths,  .00004 

Four  millionths,   -  .000004 

Four  ten  millionths,  .0000004. 

Here  we  see,  that  the  same  figure  expresses  different  deci- 
mal units,  according  to  the  place  which  it  occupies  :  therefore, 

The  value  of  the  unit,  in  the  different  places,  in  passing 
from  the  left  to  the  right,  diminishes  according  to  the  scale 
of  tens. 

Hence,  ten  of  the  units  in  any  place,  are  equal  to  one  unit  in 
the  place  next  to  the  left ;  that  is,  ten  thousandths  make  one 
hundredth,  ten  hundredths  make  one-tenth,  and  ten-tenths, 
the  unit  1. 

This  scale  of  increase,  from  the  right  hand  towards  the 
left,  is  the  same  as  that  in  whole  numbers  ;  therefore, 

Whole  numbers  and  decimal  fractions  may  be  united  by 
placing  the  decimal  point  between  them  :  thus, 

Whole  numbers.  Decimals. 


•I 

I 


836    3'0    641. 0478976 

A  number  composed  partly  of  a  whole  number  and  partly 
of  a  decimal,  is  called  a  mixed  number. 


DECIMAL   FRACTIONS.  185 


RULE    FOR   WRITING    DECIMALS. 

Write  the  decimal  as  if  it  were  a  whole  number,  prefix- 
ing as  many  ciphers  as  are  necessary  to  make  it  of  the 
required  denomination. 

RULE  FOR  READING  DECIMALS. 

Read  the  decimal  as  though  it  were  a  whole  number, 
adding  the  denomination  indicated  by  the  lowest  decimal 
unit. 

EXAMPLES. 

Write  the  following  numbers,  decimally  : 
(1.)  (2.)  (3.)  (4.)  (5.) 

3  16  17  32  165 


10  ,   1000     10000      100      10000 
(6.)      (7.)       (8.)      (9.)      (10.) 


Write  the  following  numbers  in  figures,  and  then  numerate 
them. 

1.  Forty-one,  and  three-tenths. 

2.  Sixteen,  and  three  millionths. 

3.  Five,  and  nine  hundredths. 

4.  Sixty-five,  and  fifteen  thousandths. 

5.  Eighty,  and  three  millionths. 

6.  Two,  and  three  hundred  millionths. 

7.  Four  hundred,  and  ninety-two  thousandths. 

8.  Three  thousand,  and  twenty-one  ten  thousandths. 

9.  Forty-seven,  and  twenty-one  hundred  thousandths. 

10.  Fifteen  hundred,  and  three  millionths. 

11.  Thirty-nine,  and  six  hundred  and  forty  thousandths. 

12.  Three  thousand,  eight  hundred  and  forty  millionths. 
1  3.  Six  hundred  and  fifty  thousandths. 

203.  Docs  the  value  of  the  unit  of  a  figure  depend  upon  the  place 
which  it  occupies  V  How  does  the  value  change  from  the  left  towards 
the  right  ?  What  do  ten  units  of  any  one  place  make  ?  How  do  the 
units  of  the  place  increase  from  the  right  towards  the  left  ?  How  may 
whole  numbers  be  joined  with  decimals?  What  is  such  a  number 
called?  Give  the  rule  for  writing  decimal  fractions.  Give  the  rule 
for  reading  decimal  fractions. 


186  UNITED   STATES   MONEY. 

UNITED  STATES  MONEY. 

204.  The  denominations  of  United  States  Money  correspond 
to  the  decimal  division,  if  we  regard  1  dollar  as  the  unit. 

For,  the  dimes  are  tenths  of  the  dollar,  the  cents  are  hun- 
dredths  of  the  dollar,  and  the  mills,  being  tenths  of  the  cent, 
are  thousandths  of  the  dollar. 

EXAMPLES. 

1.  Express  $39  and  39  cents  and  7  mills,  decimally. 

2.  Express  $12  and  3  mills,  decimally. 

3.  Express  $147  and  4  cents,  decimally. 

4.  Express  $148  4  mills,  decimally. 

5.  Express  $4  6  mills,  decimally. 

6.  Express  $9  6  cents  9  mills,  decimally. 

7.  Express  $10  13  cents  2  mills,  decimally. 

ANNEXING  AND  PREFIXING  CIPHERS. 

205.  Annexing  a  cipher  is   placing  it  on  the  right  of  a 
number. 

If  a  cipher  is  annexed  to  a  decimal  it  makes  one  more  deci- 
mal place,  and  therefore,  a  cipher  must  also  be  annexed  to  the 
denominator  (Art.  202). 

The  numerator  and  denominator  will  therefore  have  been 
multiplied  by  the  same  number,  and  consequently  the  value 
of  the  fraction  will  not  be  changed  (Art.  161)  :  hence, 

Annexing  ciphers  to  a  decimal  fraction  does  not  alter  its 
value. 

We  may  take  as  an  example,  .3— T37. 

If  we  annex  a  cipher,  to  the  numerator,  we  must,  at  the 
same  time,  annex  one  to  the  denominator,  which  gives, 

204.  If  the  denominations  of  Federal  Money  be  expressed  decimally 
•what  is  the  unit  ?  What  part  of  a  dollar  is  1  dime  ?  What  part  of  a 
dime  is  a  eent  ?  What  part  of  a  cent  is  a  mill  ?  What  part  of  a  dollar 
is  1  cent  ?  1  mill  ? 

305.  When  is  a  cipher  annexed  to  a  number?  Does  the  annexing 
of  ciphers  to  a  decimal  alter  its  value  ?  Why  not  ?  What  dp  three 
tenths  become  by  annexing  a  cipher  ?  What  by  annexing  two  ciphers  ? 
Three  ciphers?  What  do  8  tenths  become  by  annexing  a  cipher?  By 
annexing  two  ciphers  V  By  annexing  three  ciphers  t 


DECIMAL    FRACTIONS.  187 

,3  =      -j^j-      =  .30       by  annexing  one  cipher, 
.3  =     T3TM7°ir     —   -300     by  annexing  two  ciphers. 

if  a  decimal  point  be  placed  on  the  right  of  an  integral 
number,  and  ciphers  be  then  annexed,  the  value  will  not  be 
changed  :  thus,  5  =  5.0  =  5.00  =  5.000,  &c. 

206.  Prefixing  a   cipher   is   placing   it   on  the  left   of  a 
number. 

If  ciphers  are  prefixed  to  the  numerator  of  a  decimal  frac- 
tion, the  same  number  of  ciphers  must  be  annexed  to  the 
denominator.  Now,  the  numerator  will  remain  unchanged 
while  the  denominator  will  be  increased  ten  times  for  every 
cipher  annexed  ;  and  hence,  the  value  of  the  fraction  will  be 
diminished  ten  times  for  every  cipher  prefixed  to  the  nume- 
rator (Art.  160). 

Prefixing  ciphers  to  a  decimal  fraction  diminishes  its 
value  ten  times  for  every  cipher  prefixed. 

Take,  for  example,  the  fraction  .2=T*j-. 
.2  becomes     -ffc     =  .02      by  prefixing  one  cipher, 
.2  becomes    -fipfc    =  -002    by  prefixing  two  ciphers, 
.2  becomes  -ffiPfc  =  .0002  by  prefixing  three  ciphers  : 

in  which  the  fraction  is  diminished  ten  times  for  every  cipher 

prefixed. 

ADDITION  OF  DECIMALS. 

207.  It  must  be  remembered,  that  only  units  of  the  same 
kind  can  be  added  together.     Therefore,  in  setting  down 
decimal  numbers  for  addition,  figures  expressing  the  same 
unit  must  be  placed  in  the  same  column. 

200.  When  is  a  cipher  prefixed  to  a  number  ?  When  prefixed  to  a 
decimal,  does  it  increase  the  numerator  ?  Does  it  increase  the  denomi- 
nator? What  effect  then  has  it  on  the  value  of  the  fraction  ?  What 
do  .3  become  by  prefixing;  a  cipher?  By  prefixing  two  ciphers?  By 
prefixing  three?  What  do  .07  become  by  prefixing  a  cipher  ?  By  pre- 
fixing two  ?  By  prefixing  three  ?  By  prefixing  four  ? 

207.  What  parts  of  unity  may  be  added  together  ?  How  do  you  set 
down  the  numbers  for  addition?  How  will  the  decimal  points  fall ? 
How  do  you  then  add  ?  How  many  decimal  places  do  you  point  off  m 
the  sum  ? 


188  ADDITION  OF 

The  addition  of  decimals  is  then  made  in  the  same  manner 
•is  that  of  whole  numbers. 

I.  Find  the  sum  of  37.04,  704.3,  and  .0376. 

OPERATION. 

Place  the  decimal   points  in  the  same  column  :  HA 

this  brings  units  of  the  same  value  in  the  same  704.3 

column  :  then  add  as  in  whole  numbers  :  hence,  .0376 

741.3776 

RULE. — I.  Set  down  the  numbers  to  be  added  so  that 
figures  of  the  same  unit  value  shall  stand  in  the  same 
column. 

II.  Add  as  in  simple  numbers,  and  point  off  in  the  sum 
from  the  right  hand,  as  many  places  for  decimals  as  are  equal 
to  the  greatest  number  of  places  in  any  of  the  numbers  added. 

PROOF. — The  same  as  in  simple  numbers. 

EXAMPLES. 

1.  Add  4.035,  763.196,  445.3741,  and  91.3754  together. 

2.  Add  365.103113,  .76012,   1.34976,  .3549,  and  61.11 
together. 

3.  67.407  +  97.004+4  +  .6  +  .06  +  .3. 

4.  .0007  +  1.0436  +  .4  +  .05  +  .047. 

5.  .0049+47.0426  +  37.0410  +  360.0039. 

6.  What  is  the  sum  of  27,   14,  49,  126,  999,  .469,  and 
.2614  ? 

7.  Add  15,  100,  67,  1,  5,  33,  .467,  and  24.6  together, 

8.  What  is  the  sum  of  99,  99,  31,  .25,  60.102,  .29,  and 
100.347? 

9.  Add  together  .7509,  .0074,  69.8408,  and  .6109. 

10.  Required  the  sum  of  twenty-nine  and  3  tenths,  four 
hundred  and  sixty-five,  and   two   hundred  and   twenty-one 
thousandths. 

1 1 .  Required  the  sum  of  two  hundred  dollars  one  dime 
three  cents  and  9  mills,  four  hundred  and  forty  dollars  nine 
mills,  and  one  dollar  one  dime  and  one  mill. 

12.  What  is  the  sum  of  one-tenth,  one  hundredth,  and  one 
thousandth  ? 


DECIMAL  FRACTIONS.  189 

13.  What  is  the  sum  of  4,  and  6  ten-thousandths  ? 

14.  Required,  in  dollars  and  decimals,  the  sum  of  one  dollar 
one  dime  one  cent  one  mill,  six  dollars  three  mills,  four  dol- 
lars eight  cents,  nine  dollars  six  mills,  one  hundred  dollars  six 
dimes,  nine  dimes  one  mill,  and  eight  dollars  six  cents. 

15.  What  is  the  sum  of  4  dollars  6  cents,  9  dollars  3  mills, 
14  dollars  3  dimes  9  cents  1  mill,  104  dollars  9  dimes  9  cents 
9  mills,  999  dollars  9  dimes  1  mill,  4  mills,  6  mills,  and  1 
mill? 

16.  If  you  sell  one  piece  of  cloth  for  $4,25,  another  for 
$5,075,  and  another  for  $7,0025,  how  much  do  you  get  for 
all? 

17.  What  is  the  amount  of  $151,7,  $70,602,  $4,06,  and 
$807,2659  ? 

18.  A  man  received  at  one  time  $13,25  ;  at  another  $8,4  ; 
at  anotlier  $23,051j  at  another  $6  ;  and  at  another  $0,75  : 
how  much  did  he  receive  in  all  ? 

19.  Find  the  sum  of  twenty-five  hundredths,  three  hundred 
and  sixty-five  thousandths,  six  tenths,  and  nine  millionths. 

20.  What  is  the  sum  of  twenty-three  millions  and  ten,  one 
thousand,  four  hundred  thousandths,  twenty-seven,  nineteen 
millionths,  seven  and  five  tenths  ? 

21.  What  is  the  sum  of  six  millionths,  four  ten-thousandths, 
19  hundred  thousandths,  sixteen  hundredths,  and  four  tenths? 

22.  If  a  piece  of  cloth  cost  four  dollars  and  six  mills,  eight 
pounds  of  coffee  twenty-six  cents,  and  a  piece  of  muslin  three 
dollars  seven  dimes  and  twelve  mills,  what  will  be  the  cost 
of  them  all  ? 

23.  If  a  yoke  of  oxen  cost  one  hundred  dollars  nine  dimes 
and  nine  mills,  a  pair  of  horses  two  hundred  and  fifty  dollars 
five  dimes  and  fifteen  mills,  and  a  sleigh  sixty-five  dollars 
eleven  dimes  and  thirty-nine  mills,  what  will  be  their  entire 
cost? 

24.  Find  the  sum  of  the  following  numbers  :    Sixty-nine 
thousand  and  sixty-nine  thousandths,  forty-seven  hundred  and 
forty-seven    thousandths,    eighty-five    and    eighty-five   hun- 
dredths, six  hundred  and  forty-nine  and  six  hundred  and 
forty-nine  ten-thousandths  ? 


100  SUBTRACTION  OF 


SUBTRACTION  OF  DECIMALS 

208.  Subtraction  of  Decimal  Fractions  is  the  operation  of 
finding  the  difference  between  two  decimal  numbers. 

I.  From  3.275  to  take  .0879. 

NOTE. — In   this  example   a  cipher  is   annexed  OPSBATION. 
to  the  minuend  to  make  the   number  of  decimal         3.2750 
places  equal  to  the  number  in  the  subtrahend.     This  08  *7  Q 

does  not  alter  the  value  of  the  minuend  (Art.  205)  • 
hence,  3.1871 

RULE. — I.   Write  the  less  number  under  the  greater,  so  that 
figures  of  the  same  unit  value  shall  stand  in  the  same  column. 

II.  Subtract  as  in  simple  numbers,  and  point  off  the  deci- 
mal places  in  the  remainder,  as  in  addition. 

PROOF. — Same  as  in  simple  numbers. 

EXAMPLES. 

1.  From  3295  take  .0879. 

2.  From  291.10001  take  41.375. 

3.  From  10.000001  take  111111. 

4.  From  396  take  8  ten-thousandths. 

5.  From  1  take  one  thousandth. 

6.  Fcom  6378  take  one-tenth. 

7.  From  365.0075  take  3  millionths. 

8.  From  21.004  take  97  ten-thousandths. 

9.  From  260.4709  take  47  ten-millionths. 

10.  From  10.0302  take  19  millionths. 

11.  From  2.01  take  6  ten-thousandths. 

12.  From  thirty-five  thousands  take  thirty-fire  thousandths. 

13.  From  4262.0246  take  23.41653. 

14.  From  346.523120  take  219.691245943. 
'     15.  From  64.075  take  .195326. 

16.  What  is  the  difference  between  107  and  .0007? 

17.  What  is  the  difference  between  1.5  and  .3785  ? 

18.  From  96. 71  take  96.709. 


208.  What  is  subtraction  of  decimal  fractions  ?  How  do  you  set  down 
the  numbers  for  subtraction  ?  How  do  you  then  subtract  ?  How  many 
decimal  places  do  you  point  off  in  the  remainder  ? 


DECIMAL   FRACTIONS.  191 

MULTIPLICATION  OF  DECIMAL  FRACTIONS. 

209.  To  multiply  one  decimal  by  another. 
1.  Multiply  3.05  by  4.102. 

OPERATION. 

ANALYSIS. — If  we  change  both  factors  to  vul-  s. «£  —  3  05 

£&r  fractions,  the  product  of  the  numerator  will  4JJL2.— 1   1Q9 
be  the  same  as  that  of  the  decimal  numbers,  and 

the  number  of  decimal  places  will  be  equal  to  the  610 

number  of  ciphers  in  the  two  denominators:  305 

hence,  12 . 20 

12.51110 

RULE. — Multiply  as  in  simple  numbers,  and  point  off"  in 
the  product,  from  the  right  hand,  as  many  figures  for  decimals 
as  there'are  decimal  places  in  both  factors  ;  and  if  there  be 
not  so  many  in  the  product,  supply  the  deficiency  by  prefixing 
ciphers. 

EXAMPLES 

1.  Multiply  3. 049  by  .012. 

2.  Multiply  365.491  by  .001. 

3.  Multiply  496. 0135  by  1.496. 

4.  Multiply  one  and  one  milliouth  by  one  thousandth. 

5.  Multiply  one  hundred  and  forty-seven  millionths  by  one 
millionth. 

6.  Multiply  three  hundred,  and  twenty-seven  hundredth^ 
by  31. 

7.  Multiply  31.00467  by  10.03962. 

8.  What  is  the  product  of  five-tenths  by  five-tenths  ? 

9.  What  is  the  product  of  five-tenths  by  five-thousandths  ? 

10.  Multiply  596.04  by  0.00004. 

11.  Multiply  38049.079  by  0.00008. 

12.  What  will  6.29  weeks'  board  come  to  at  2.75  dollars 
per  week  ? 

13.  What  will  61  pounds  of  sugar  come  to  at  $0.234  per 
pound  ? 

209.  After  multiplying,  how  many  decimal  places  will  you  point  off 
In  the  product  ?  When  there  are  not  so  many  in  the  product  what  do 
you  do  ?  Give  the  rule  for  the  multiplication  of  decimals. 


192 


CONTRACTIONS. 


14.  If  12  .  836  dollars  are  paid  for  one  barrel  of  flour,  what 
will  .  354  barrels  cost  ? 

15.  What  are  the  contents  of  a  board,  .  06  feet  long  and  .  06 
wide? 

16.  Multiply  49000  by  .0049. 

17.  Bought  1234  oranges  for  4  .  6  cents  apiece  :  how  much 
did  they  cost  ? 

18.  What  will  375.6  pounds  of  coffee  cost  at  .125  dollars 
per  pound  ? 

19.  If  I  buy  36.  251  pounds  of  indigo  at  $0.029  per  pound, 
what  will  it  come  to  ? 

20.  Multiply  $89.  3421001  by  .0000028. 

21.  Multiply  $341.45  by  .007. 

22.  What  are  the  contents  of  a  lot  which  is  .  004  miles  long 
and  .  004  miles  wide  ? 

23.  Multiply  .007853  by  .035. 

24.  What  is  the  product  of  $26.000375  multiplied  1>v 
.00007? 


CONTRACTIONS. 

210.  When  a  decimal  number  is  to  be  multiplied  by  10, 
100,  1000,  &c.,  the  multiplication  may  be  made  by  removing 
the  decimal  point  as  many  places  to  the  right  hand  as  there 
are  ciphers  in  the  multiplier,  and  if  there  be  not  so  many 
figures  on  the  right  of  the  decimal  point,  supply  the  deficiency 
by  annexing  ciphers. 


Thus,  6.79  multiplied  by    - 


10 

100 
1000 
10000 
100000 


Also,  370 . 036  multiplied  by 


flO         1 

|100 

1000     L  •= 

10000 
100000  J 


67.9 

679 

6790 

67900 

679000 

3700.36 
37003.6 
370036 
3700360 
37003600 


210.  How  do  you  multiply  a  decimal  number  by  10,  100,  1000,  Ac.  ? 
If  there  are  not  as  many  decimal  figures  as  there  are  ciphers  in  the 
multiplier,  what  do  you  <lo  ? 


DECIMAL   FRACTIONS.  193 

DIVISION  OF  DECIMAL  FRACTIONS. 

211.  Division  of  Decimal  Fractions  is  similar  to  that  of 
simple  numbers. 

1.  Let  it  be  required  to  divide  1.38483  by  60.21. 

ANALYSIS. — The  dividend  must  be  equal  OPERATION. 

to  the  product  of  the  divisor  and  quotient,      60 . 21 )  1 . 38483(23 
(Art,   61) ;    and    hence    must    contain   as  j   2042 

many   decimal  places  as   both  of  them ; 
therefore, 

There,  must  be  as  many  decimal  places  in  18063 

the  quotient  as  the  decimal  places  in  the  divi-  ~r~     7\wi 

dend  exceed  those  in  the  divisor  :  hence, 

R.ULE. — Divide  as  in  simple  numbers,  and  point  off"  in  the 
quotient,  from  the  right  hand,  as  many  places  for  decimals  as 
the  decimal  places  in  the  dividend  exceed  those  in  the  divisor  ; 
and  if  there  are  not  so  many,  supply  the  deficiency  by  prefix- 
ing ciphers. 


EXAMPLES. 


1.  Divide  2.3421  by  2.11 

2.  Divide  12.82561  by  3.01. 

3.  Divide  33.66431  by  1.01. 


4.  Divide  .010001  by  .01. 

5.  Divide  8.2470  by  .002. 

6.  Divide  94.0056  by  .08. 


7.  What  is  the  quotient  of  37  . 57602,  divided  by  3  ;  by  .  3  ; 
by  .03;  by  .003;  by  .0003? 

8.  What  is  the  quotient  of  129.75896,  divided  by  8  ;  by 
.08;  by  .008;  by  .0008;  by  .00008? 

9.  What  is  the  quotient  of  187  .29900,  divided  by  9  ;  by 
.9  ;  by  .09  ;  by  .009  ;  by  .0009  ;  by  .00009  ? 

10.  What  is  the  quotient  of  764  2043244,  divided  by  6  ; 
by  .06  ;  by  .006  ;  by  .0006  ;  by  .00006  ;  by  .000006? 

NOTE. — 1.  When  there  are  more  decimal  places  in  the  divisor 
than  in  the  dividend,  annex  ciphers  to  the  dividend  and  make  the 
decimal  places  equal ;  all  the  figures  of  the  quotient  will  then  be 
whole  numbers. 


211.  How  docs  the  number  of  decimal  places  in  the  dividend  com- 
pare with  that  in  the  divisor  and  quotient?  How  do  you  determine 
the  number  of  decimal  places  in  the  quotient?  If  the  divisor  contains 
four  places  and  the  dividend  six,  how  many  in  the  quotient  ?  If  the 
divisor  contains  three  places  and  the  dividend  five,  how  many  in  the 
quotient  ?  Give  the  rule  for  the  division  of  decimals. 
13 


DIVISION  OF 


EXAMPLES. 


1.  Divide  4397. 4  by  3. 49. 


NOTE. — We  annex  one  0  to 
the  dividend.  Had  it  contained 
no  decimal  place  we  should 
have  annexed  two. 


OPERATION. 
3.49)4397.40(1260 
349 

907 
698 


2094 
2094 


An*.   1260. 


2.  Divide  2194.02194  by  .100001. 

3.  Divide  9811. 0047  by  .325947. 

4.  Divide  .1  by  .0001.         |      5.  Divide  10  by  .15. 

6.  Divide  6  by  .6  ;  by  .06  ;  by. 006  ;  by  .2  ;  by  .3  ;  by 
.003;  by  .5;  by  .05;  by  .005. 

NOTE. — 2.  When  it  is  necessary  to  continue  the  division  farther 
than  the  figures  of  the  dividend  will  allow,  we  annex  ciphers,  and 
consider  them  as  decimal  places  of  the  dividend. 

When  the  division  does  not  terminate,  we  annex  the  plus  sign 
to  show  that  it  may  be  continued :  thus  .2  divided  by  ^=.666+. 


EXAMPLES. 


1.  Divide  4. 25  by  1.25. 

ANALYSIS. — In  this  example  we  annex  one  0. 
and  then  the  decimal  places  in  the  dividend  will 
exceed  those  in  the  divisor  by  1. 


OPERATION. 

25)4.25(3.4 

3.75 

~500 

500 

Ans.  3.4. 


2.  Divide  .  2  by  .6. 

3.  Divide  37. 4  by  4. 5. 


4.  Divide  586.4  by  375. 

5.  Divide  94 . 0369  by  81 . 032. 


NOTE. — 3.  When  any  decimal  number  is  to  be  divided  by  10, 
100,  1000,  &c.,  the  division  is  made  by  removing  the  decimal 
point  as  many  places  to  the  left  as  there  are  Q's  in  Vie  divisor ;  and 
if  there  be  not  so  many  figures  on  the  left  of  the  decimal  point, 
the  deficiency  is  supplied  by  prefixing  ciphers. 


27 . 69  divided  by 


10 
100 
1000 
10000 


2.769 
.2769 
.02769 
.002769 


DECIMAL  FRACTIONS.  195 


10 

100 

642.89  divided  by  -I  1000 
10000 
100000 


64.289 
6.4289 
.64289 
.064289 
.0064289 


QUESTIONS    IN   THE    PRECEDING    RULES 

1.  If  I  divide  .6  dollars  among  94  men,  how  much  will 
each  receive  ? 

2.  I  gave  28  dollars  to  267  persons  :  how  much  apiece  ? 

3.  Divide  6  35  by  .425. 

4.  What  is  the  quotient  of  $36.2678  divided  by  2.25  ? 

5.  Divide  a  dollar  into  12  equal  parts. 

6.  Divide  .25  of  3.26  into  .034  of  3.04  equal  parts. 

7.  How  many  times  will  .35  of  35  be  contained  in  .024 
of  24?   * 

8.  At  .75  dollars  a  bushel,  how  many  bushels  of  rye  can 
be  bought  for  141  dollars  ? 

9.  Bought  12  arid  15  thousandths  bushels  of  potatoes  for 
33  hundredths  dollars  a  bushel,  and  paid  in  oats  at  22  hun- 
dredths  of  a  dollar  a  bushel :  how  many  bushels  of  oats  did  it 
take? 

10.  Bought  53.1  yards  of  cloth  for  42  dollars  :  how  much 
was  it  a  yard  ? 

11.  Divide  125  by  .1045. 

12.  Divide  one  millionth  by  one  billionth. 

1 3.  A  merchant  sold  4  parcels  of  cloth,  the  first  contained 
127  and  3  thousandths  yards  ;  the  2d,  6  and  3  tenths  yards  ; 
the  3d,  4  and  one  hundredth  yards  ;  the  4th,  90  and  one 
millionth  yards  :  how  many  yards  did  he  sell  in  all  ? 

14.  A  merchant  buys  three  chests  of  tea,  the  first  contains 
60  and  one  thousandth  pounds  ;  the  second,  39  and  one  ten 
thousandth  pounds  ;  the  third,  26  and  one  tenth  pounds  :  how 
much  did  he  buy  in  all  ? 

NOTE.— 1.  If  there  are  more  decimal  places  in  the  divisor  than  in  the 
dividend,  what  do  you  do  ?  What  will  the  figures  of  the  quotient  then 
be? 

2.  How  do  you  continue  the  division  after  you  have  brought  down  all 
the  figures  of  the  dividend  ?    What  sign  do  you  place  after  the  quo- 
tient ?    What  does  it  show? 

3.  How  do  you  divide  a  decimal  fraction  by  10,  100, 1000,  &c.  ? 


19G  DIVISION   OF 

15.  What  is  the  sum  of  $20  and  three  hundredths  ;  $4 
and  one-tenth,   $6  and  one   thousandth,   and  $18  and  one 
hundredth  ? 

16.  A  puts  in  trade  $504.342  ;  B  puts  in  $350.1965  ;  C 
puts   in   $100.11;     D    puts   in   $99.334;    and   E   puts   in 
$9001.32  :  what  is  the  whole  amount  put  in  ? 

It.  B  has  $936,  and  A  has  $1,  3  dimes  and  1  mill :  how 
much  more  money  has  B  than  A  ? 

18.  A  merchant  buys  37.5  yards  of  cloth,  at  one  dollar 
twenty-five   cents   per    yard :    how   much  does   the   whole 
come  to  ? 

19.  If  12  men  had  each  $339  one  dime  9  cents  and  3 
mills,  what  would  be  the  total  amount  of  their  money  ? 

20.  A  farmer  sells  to  a  merchant  13.12  cords  of  wood  at 
$4.25  per  cord,  and  13  bushels  of  wheat  at  $1.06  per  bushel : 
he  is  to  take  in  payment  13  yards  of  broadcloth  at  $4.07  per 
yard,  and  the  remainder  in  cash :  how  much  money  did  he 
receive  ? 

21.  If  one  man  can  remove  5.91  cubic  yards  of  earth  in  a 
day,  how  much  could  nineteen  men  remove  ? 

22.  What  is  the  cost  of  8.3  yards  of  cloth  at  $5.47  per 
yard? 

23.  If  a  man  earns  one  dollar  and  one  mill  per  day,  how 
much  will  he  earn  in  a  year  of  313  working  days  ? 

24.  What  will  be  the  cost  of  375  thousandths  of  a  cord  of 
wood,  at  $2  per  cord  ? 

25.  A  man  leaves  an  estate  of  $1473.194  to  be  equally 
divided  among  12  heirs  :  what  is  each  one's  portion  ? 

26.  If  flour  is  $9.25  a  barrel,  how  many  barrels  can  I  buy 
for  $1637.25  ? 

27.  Bought  26  yards  of  cloth  at  $4.37|  a  yard,  and  paid 
for  it  in  flour  at  $7.25  a  barrel :  how  much  flour  will  pay 
for  the  cloth  ? 

28.  How  much  molasses  at  22|-  cents  a  gallon  "must  be 
given  for  46  bushels  of  oats  at  45  cents  a  bushel? 

29.  How  many  days  work  at  $1.25  a  day  must  be  given 
for  6  cords  of  wood,  worth  $4.12|  a  cord? 

30  What  will  36.48  yards  of  cloth  cost,  if  14.25  yards 
cost  $21. 375? 

31.  If  you  can  buy  13.25/6.  of  coffee  for  $2.50,  how  much 
can  you  buy  for  $325.50  ? 


DECIMAL   FRACTIONS. 


197 


212.  To  change  a  common  to  a  decimal  fraction. 

The  value  of  a  fraction  is  the  quotient  of  the  numerate! 
divided  by  the  denominator  (Art.  148). 

1.  Reduce  J  to  a  decimal. 

If  we  place  a  decimal  point  after  the  5,  and  then  OPERATION. 
write  any  number  of  O's,  after  it,  the  value  of  the  8)5.000 
numerator  will  not  be  changed  (Art.  205).  T'9f\ 

If,  then,  we  divide  by  the  denominator,  the  quo- 
tient will  be  the  decimal  number  :  hence, 

RULE.  —  Annex  decimal  ciphers  to  the  numerator,  and 
then  divide  by  the  denominator,  pointing  off  as  in  division 
of  decimals. 


1  .  tteduce 


EXAMPLES. 

to  its  equivalent  decimal. 


We  here  use  two  ciphers,  and  therefore  point 
off  two  decimal  places  in  the  quotient, 


Reduce  the  following"  fractions  to  decimals 


OPERATION. 

125)635(5.08 
625 
1000 
1000 


to  a  decimal. 


1.  Reduce  -^  to  a  decimal. 

2.  Reduce  -J-f-  to  a  decimal. 

3.  Reduce  -fa  to  a  decimal. 

4.  Reduce  J  and  t  ^  5 . 

5.  Reduce  ^5-,  f  f ,  and 

6.  Reduce  J  and 
V.  Reduce 

8.  Reduce  f, 

9.  Reduce  ££  to  a  decimal. 

213.  A  decimal  fraction  may  be  changed  to  the  form  of  a 
vulgar  fraction  by  simply  writing  its  denominator  (Art.  202). 

212.  How  do  you  change  a  vulgar  to  a  decimal  fraction  ? 

213.  How  do  you  change  a  decimal  to  the  form  of  a  vulgar  fraction  ? 


10.  Reduce 

11.  Reduce 

12.  Reduce 

13.  Reduce 

14.  Reduce  T 

15.  Reduce 

16.  Reduce 

17.  Reduce 

18.  Reduce 


198  DENOMINATE  DECIMALS. 

EXAMPLES. 

1.  What  vulgar  fraction  is  equal  to  .04  ? 

2.  What  vulgar  fraction  is  equal  to  3.067  ? 

3.  What  vulgar  fraction  is  equal  to  8.275  ? 

4.  What  vulgar  fraction  is  equal  to  .00049  ? 

DENOMINATE  DECIMALS. 

214.  A  denominate  decimal  is  one  in  which  the  unit  of  the 
fraction  is  a  denominate  number.     Thus,  .5  of  a  pound,  .6  of  a 
shilling,  .7  of  a  yard,  &c.,  are  denominate  decimals,  in  which 
the  units  are  1  pound,  1  shilling,  1  yard. 

CASE    I. 

215.  To  change  a  denominate  number  to  a  denominate 
decimal. 

1.  Change  9tf.  to  the  decimal  of  a  £. 

ANALYSIS.— The  denominate  unit  of  the  frac-  OPERATION. 

tion  is    l£=24Qd.      Then  divide  Qd.   by  240:  2±Qd.=£l 

the  quotient,   .0375  of  a  pound  is  the  value  of  240) 9 (.03 7 5 

9dJ.  in  the  decimal  of  a  £  :  hence,  ^     .  ^  0375 

RULE. — Reduce  the  unit  of  the  required  fraction  to  the  unit 
of  the  given  denominate  number,  and  then  divide  the  denomi- 
nate number  by  the  result,  and  the  quotient  will  be  the  decimal. 

EXAMPLES. 

1.  Reduce  7  drams  to  the  decimal  of  a  Ib.  avoirdupois. 

2.  Reduce  26d.  to  the  decimal  of  a  £. 

3.  Reduce  .056  poles  to  the  decimal  of  an  acre. 

4.  Reduce  14  minutes  to  the  decimal  of  a  day. 

5.  Reduce  21  pints  to  the  decimal  of  a  peck. 

6.  Reduce  3  hours  to  the  decimal  of  a  day. 

7.  Reduce  375678  feet  to  the  decimal  of  a  mile. 

8.  Reduce  36  yards  to  the  decimal  of  a  rod. 

9.  Reduce  .5  quarts  to  the  decimal  of  a  barrel. 

10.  Reduce  .7  of  an  ounce,  avoirdupois,  to  the  decimal  of  a 
hundred. 

214.  What  is  a  denominate  decimal  ? 

215.  How  do  you  change  a  denominate   number  to  a  denominate 
decimal  ? 


DENOMINATE   DECIMALS.  199 

CASE    II. 

216.  To  find  the  value  of  a  decimal  in  integers  of  a  less 
denomination. 

1.  Find  the  value  of  .890.625  bushels. 

OPERATION. 

ANALYSIS.  —  Multiplying  the  decimal  by  4,  (since  4  890625 

pecks  make  a  bushel),  we  have  3,5625  pecks.     Mul-  \ 

tiplying  the  new  decimal  by  8,  (since  8  quarts  make  __  _ 

a  peck),  we  have  4.5   quarts.     Then,  multiplying  3.562500 

this  last  decimal  by  2,  (since  2  pints  make  a  quart),  8 

we  have  1  pint;  hence,  4.500000 

2 


_ 

.  Bpk.  Iqts.  Ipt.     1.000000 

RULE.  —  I.  Multiply  the  decimal  by  that  number  which 
will  reduce  it  to  the  next  less  denomination,  pointing  off  as 
in  multiplication  of  decimal  fractions. 

1  1  .  Multiply  the  decimal  pa  rt  of  the  product  as  before  ;  and 
so  continue  to  do  until  the  decimal  is  reduced  to  the  required 
denominations.  The  integers  at  the  left  form  the  answer. 

EXAMPLES. 

1.  What  is  the  value  of  .002084/6.  Troy? 

2.  What  is  the  value  of  .  625  of  a  cwt.  ? 

3.  What  is  the  value  of  .  625  of  a  gallon  ? 

4.  What  is  the  value  of  £  .  3375  ? 

5.  What  is  the  value  of  .  3375  of  a  ton  ? 

6.  What  is  the  value  of  .  05  of  an  acre  ? 

7.  What  is  the  value  of  .  875  pipes  of  wine  ? 

8.  What  is  the  value  of  .125  hogshead   of  beer  ? 

9.  What  is  the  value  of  .  375  of  a  year  of  365  days  ? 

10.  What  is  the  value  of  .  085  of  a  £  ? 

11.  What  is  the  value  of  .86  of  a  cwt.  ? 

12.  From  .82  of  a  day  take  .32  of  an  hour. 

13.  What  is  the  value  of  1.089  miles? 

14.  What  is  the  value  of  .09375  of  a  pound,  avoirdupois  ? 

15.  What  is  the  value  of  .28493  of  a  year  of  365  days  ? 

16.  What  is  the  value  of  £1.046? 

17.  What  is  the  value  of  £1.88  ? 


216.  How  do  you  find  the  value  of  a  decimal  in  integers  of  a  less 
denomination  ? 


200  DENOMINATE   DECIMALS 


CASE    III. 

217.  To  reduce  a  compound  denominate  number  to  a 
decimal  or  mixed  number. 

1.  Reduce  £1  4s.  9|c?.  to  the  decimal  of  a  £. 

ANALYSIS.— Reducing  the  f<f.  to  a  decimal 
(Art.  215),  and  annexing  the  result  to  the  9d,  *  ?  _  I  ^ 

we  have  9.75d.     Dividing  9 .75d.  by  12,  (since          $T~~  ' 
12  pence— Is.),  and  annexing  the  quotient  to        y$d.  =  $  .*l5d. 
the  4s.  we  have  4.8125s.    Then,  dividing  by  20        12)9   75c? 
(since  20s.=£l,)  and  annexing  the  quotient 
to  the  £1,  we  have  £1.240625  : 

Ans.  £1  4s.  9|d.  =  1.240625£. 

RULE. — Divide  the  lowest  denomination  by  as  many  units 
as  make  a  unit  of  the  next  higher,  and  annex  the  quotient 
as  a  decimal  to  that  higher:  then  divide  as  before,  and  so 
continue  to  do  until  the  decimal  is  reduced  to  the  required 
denomination. 

EXAMPLES. 

1.  Reduce  kwk.  $da.  5/ir.  30m.  45s,  to  the  denomination 
of  a  week. 

2.  Reduce  2/6.  5oz.  I2pwt.  Iftgr.,  to  the  denomination  of  a 
pound. 

3.  Reduce  3  feet  9  inches  to  the  denomination  of  yards. 

4.  Reduce  1/6.  12dr.,  avoirdupois,  to  the  denomination  of 
pounds. 

5.  Reduce  5  leagues  2  furlongs  to  the  denomination  of 
leagues. 

6.  Reduce  46u.  %pk.  4=qt.   Ipt.  to    the    denomination  of 
bushels. 

7.  Reduce  5oz.  ISpwt.  I2gr.  to  the  decimal  of  a  pound. 

8.  Reduce  Ibcwt.  3qr.  2J/6.  to  the  decimal  of  a  ton. 

9.  Reduce  5A  3/?.  21sg.  rd.  to  the  denomination  of  acres. 

10.  Reduce  11  pounds  to  the  decimal  of  a  ton. 

1 1.  Reduce  3efa.  l%%xcc.  to  the  decimal  of  a  week. 

12.  Reduce  146w.  3%qt.  to  the  decimal  of  a  chaldron. 

13.  Reduce  7m.  7/wr.  Ir.  to  the  denomination  of  miles. 


217.  How   do   you   reduce   a   compound    denominate    number   to 
a  decimal  V 


ANALYSIS.  201 


ANALYSIS. 

218.  An  analysis  of  a  proposition  is  an  examination  of  its 
separate  parts,  and  their  connections  with  each  other. 

The  solution  of  a  question,  by  analysis,  consists  in  an  exami- 
nation of  its  elements  and  of  the  relations  which  exist  between 
these  elements.  We  determine  the  elements  and  the  rela- 
tions which  exist  between  them,  in  each  case,  by  examining 
the  nature  of  the  question. 

In  analyzing,  we  reason  from  a  given  number  to  its  unit, 
and  then  from  this  unit  to  the  required  number. 

EXAMPLES. 

1.  If  9  bushels  of  wheat  cost  18  dollars,  what  will  21 
bushels  cost  ? 

ANALYSIS. — One  bushel  of  wheat  will  cost  one  ninth  as  much  as 
9  bushels.  Since  9  bushels  cost  18  dollars,  1  bushel  will  cost  ^ 
of  18  dollars,  or  2  dollars;  27  bushels  will  cost  27  times  as  much 
as  1  bushel :  that  is,  27  times  ^  of  18  dollars  or  54  dollars. 


OPERATION. 

0       18 

O>7  V         <*•*• 

—=$54  ;     Or, 


' 


|  54  .4ns. 

NOTE. — 1.  We  indicate  the  operations  to  be  performed,  and 
then  cancel  the  equal  factors  (Art.  141). 

219.  Although  the  currency  of  the  United  States  is  ex- 
pressed in  dollars  cents  and  mills,  still  in  most  of  the  States 
the  dollar  (always  valued  at  100  cents),  is  reckoned  in  shil- 
lings and  pence  ;  thus, 

In  the  New  England  States,  in  Indiana,  Illinois,  Missouri,  Vir 
ginia.  Kentucky,  Tennessee,  Mississippi  and  Texas,  the  dollar  is 
reckoned  at  G  shillings:  In  New  York,  Ohio  and  Michigan,  at  8 
shillings:  In  New  Jersey,  Pennsylvania,  Delaware  and  Mary 
land,  at  7s.  6d.  :  In  South  Carolina,  and  Georgia,  at  4s.  8d. :  In 
Canada  and  Nova  Scotia,  at  5  shillings. 

21S.  What  is  an  analysis  ?  In  what  does  the  solution  of  a  question 
by  analysis  consist  ?  How  do  we  determine  the  elements  and  their 
relations  ?  How  do  we  reason  in  analyzing  V 


202 


ANALYSIS. 


NOTE  —  In  many  of  the  States  the  retail  price  of  articles  is  given 
in  shillings  and  pence,  and  the  result,  or  cost,  required  in  dollars 
and  cents. 

2.  What  will  12  yards  of  cloth  cost,  at  5  shillings  a  yard, 
New  York  currency  ? 

ANALYSIS.—  Since  1  yard  cost  5  shillings  12  yards  will  cost  12 
times  5  shillings,  or  60  shillings  and  as  8  shillings  make  1  dollar, 
New  York  currency,  there  will  be  as  many  dollars  as  8  is  contain- 
ed timesin60=$7ir. 


OPERATION. 


5xl2-^8=$7.50;      Or, 


n 

5 


2  |  15  =  ^=$7.50. 
$!.50. 

NOTE. — The  fractional  part  of  a  dollar  may  always  be  reduced 
to  cents  and  mills  by  annexing  two  or  three  ciphers  to  the  nume- 
rator and  dividing  by  the  denominator ;  or,  which  is  more  conve- 
nient in  practice,  annex  the  ciphers  to  the  dividend  and  continue 
the  division. 

3.  What  will  be  the  cost  of  56  bushels  of  oats  at  3s  Zd  a 
bushel,  New  York  currency  ? 

OPERATION. 


Or, 


4  |  91 

$22.75  Am. 


NOTE.  —  When  the  pence  is  an  aliquot  part  of  a  shilling  the 
price  may  be  reduced  to  an  improper  fraction,  which  will  be  the 
multiplier:  thus,  8l  8d.=8i*.=1/«.  Or:  the  shillings  and  pence 
may  be  reduced  to  pence;  thus,  3s  3d.  ~39rf.,  in  which  case  the, 
product  will  be  pence,  and  must  be  divided  by  96,  the  number  of 
pence  in  1  dollar  :  hence, 

220.   To  find  the  cost  of  articles  in  dollars  and  cents. 


219;  In  what  is  the  currency  of  the  States  expressed  ? 
the  currency  of  the  States  often  reckoned  ? 
220.  How  do  you  find  the  cost  of  a  commodity  ? 


In  what  is 


ANALYSIS.  203 

Multiply  the  commodity  by  the  price  and  divide  theprodutc 
by  the  value  of  a  dollar  reduced  to  the  same  denominational 
unit. 

4.  What  will  18  yards  of  satinet  cost  at  3s.  §d.  a  yard, 
Pennsylvania  currency  ? 

OPERATION. 


Or,  *  00 


\  $y.  |  $9  Ans. 

NOTE. — The  above  rule  will  apply  to  the  currency  in  any  of 
the  States.  In  the  last  example  the  multiplier  is  3s.  9c?.=3J*. 
=J£*.  or  46d.  The  divisor  is  7*.  W.=7|*.=^f.=90tl, 

5.  What  will  7J/6.  of  tea  cost  at  6s.  Sd.  a  pound,  New 
Englan4  currency  ?  • 


OPERATION. 

t 

t$  L  5 

**n 

*}* 

3* 

20  l                     Or, 

00   ™ 

0 

3 

25 

6.  What  will  be  the  cost  of  120?/^s.  of  cotton  cloth  at  Is. 
f)d.  a  yard,  Georgia  currency  ? 

7.  What  will  be  the  cost  in  New  York  currency  ? 

8.  What  will  be  the  cost  in  New  England  currency  ? 

9.  What  will  be  the  cost  of  75  bushels  of  potatoes  at  3s. 
6d.,  New  York  currency  ? 

10.  What  will  it  cost  to  build  148  feet  of  wall  at  Is.  Sd. 
per  foot,  N.  Y.  currency  ? 

11.  What  will  a  load  of  wheat,  containing  46 J  bushels 
come  to  at  10s.  Sd.  a  bushel,  N.  Y.  currency? 

12.  What  will  7  yards  of  Irish  linen  cost  at  3s.  4d.  a  yard, 
Pcnn.  currency  ? 

13.  Kow  many  pounds  of  butter  at  Is.  4d.  a  pound  must 
be  given  for  12  gallons  of  molasses  at  2s.  Sd.  a  gallon  ? 


204 


ANALYSIS. 


12 


OPERATION. 

Or, 


12 


24/6. 


|  24/6. 

NOTE. — The  same  rule  applies  in  the  last  example  as  in  the 
preceding  ones,  except  that  the  divisor  is  the  price  of  the  article 
received  in  payment,  reduced  to  the  same  unit  as  the  price  of  the 
article  bought. 

14.  What  will  be  the  cost  of  12cwt.  of  sugar  at  9cZ.  per  /&. 
N.  Y.  currency? 


OPERATION. 

0 

25 
9 


2     225 


NOTE.— Reduce  the  cicts.  to  Ibs.  by 
multiplying  by  4  and  then  by  25.  Then  2  ^ 
multiply  by  the  price  per  pound,  and 
then  divide  by  the  value  of  a  dollar  in 
the  required  currency,  reduced  to  the 
same  denomination  asjthe  price. 

Ans.  $112,50 

15.  What  will  be  the  cost  of  9  hogsheads  of  molasses  at  Is. 
3d.  per  quart,  N.  E.  currency  ? 

16.  How  many  days  work  at  7s.  6c?.  a  day  must  be  given 
for  1 2  bushels  of  apples  at  3s.  $d.  a  bushel  ? 

17.  Farmer  A  exchanged  35  bushels  of  barley,  worth  6s. 
4d.t  with  farmer  B  for  rye  worth  7  shillings  a  bushel  :  how 
many  bushels  of  rye  did  farmer  A  receive  ? 

18.  Bought  the  following  bill  of  goods  of  Mr.  Merchant : 
what  did  the  whole  amount  to,  N.  Y.  currency  ? 

12|  yards  of  cambric         at  Is. 

8       "  ribbon 

21       "          calico 

6       "  alpaca 

4    gallons  molasses 

2J  pounds  tea 
30         "       sugar 

19.  Iff  of  a  yard  of  cloth  cost  $3.20,  what  will  -}-£  of  a 
yard  cost  ? 

ANALYSIS. — Since  5  eighths  of  a  yard  of  cloth  costs  $3,20, 1  eighth 
of  a  yard  will  cost  i  of  $3,20  ;  and  1  yard,  or  8  eighths,  will  cost 
8  times  as  muck,  or  £  of  $3,20,  |$  of  a  yard  will  cost  i£  as  much 
as  1  yard,  or  i$  of  £  of  $3.20= $4.80. 


4d  per  yard. 
2s.  6d.       " 
Is.  3d.       " 
5s.  Qd,       " 
3s.  bd.  per  gallon. 
6s.  6c?.  per  pound. 


ANALYSIS.  205 

OPERATION. 


1.60        ,        *       yi 

*.20xlx?xi?=$4.SO.     Or, 

&       1       40 


$4.80. 


20.  If  3  j  pounds  of  tea  cost  3^  dollars,  what  will  9  pounds 
cost? 

NOTE. — Reduce  the  mixed  numbers  to  improper  fractions,  and 
then  apply  the  same  mode  of  reasoning  as  in  the  preceding  ex- 
ample. 

21.  What  will  8|  cords  of  wood  cost,  if  2f  cords  cost  7J- 
dollars  ? 

22.  If  6  men  can  build  a  boat  in  120  days,  how  long  will 
it  take  24  men  to  build  it  ? 

ANALYSIS. — Since  6  men  can  build  .a  boat  in  120  days,  it  will 
take  1  man  6  times  120  days,  or  720  days,  and  24  men  can  build 
it  in  fa  of  the  time  that  1  man  will  require  to  build  it,  or  fa  of  G 
times  120,  which  is  30 

OPERATION. 

30 
120x6 -=-24  =  30  days.     Or,     £ 

M 


Ans.      30  days, 

23  If  7  men  can  dig  a  ditch  in  21  days,  how  many  men 
will  be  required  to  dig  it  in  3  days  ? 

24.  In  what  time  will  12  horses  consume  a  bin  of  oats, 
that  will  last  21  horses  6f  weeks  ? 

25.  A  merchant  bought  a  number  of  bales  of  velvet,  each 
containing  129^  yards,  at  the  rate  of  7  dollars  for  5  yards, 
and  sold  them  at  the  rate  of  1 1  dollars  for  7  yards  ;  and 
gained   200  dollars  by  the  bargain  :    how  many  bales  were 
there  ? 

ANALYSTS  — Since  he  paid  7  dollars  for  5  yards,  for  1  yard  he 
paid  ^  of  $7  or  I  of  1  dollar  ;  and  since  he  received  11  dollars  for 
7  yards,  for  1  yard  he  received  |  of  11  dollars  or  V-  of  1  dollar 
He  gained  on  1  yard  the  difference  between  £  and  V~=-35«r  of  a  dol 
lar.  Since  his  whole  gain  was  200  dollars,  he  had  as  many  yards 
as  the  gain  on  one  yard  is  contained  times  in  his  whole  gain,  or 
as  :ft,  is  contained  times  in  200.  And  there  were  as  many  bales 
as  129 1^,  (the  number  of  yards  in  one  bale),  is  contained  times  in 
the  whole  number  of  yards  ^^ ;  which  gives  9  bales. 


206  ANALYSIS. 


OPERATION. 

=  3500,  number  of  yards  in  a  bale  :          * 

<& 

-=-^65=-2-%0-0-,  whole  number  of  yards:  ^00 

LAO_0— -9  K~l™  $$00 


200 


* 


26.  Suppose  a  number  of  bales  of  cloth,  each  containing 
133^  yards,  to  be  bought  at  the  rate  of  12  yards  for  11  dol- 
lars, and  sold  at  the  rate  of  8  yards  for  7  dollars,  and  the 
loss  in  trade  to  be  $100  :  how  many  bales  are  there  ? 

27.  If  a  piece  of  cloth  9  feet  long  and  3  feet  wide,  contain 
3  square  yards  ;  how  long  must  a  piece  of  cloth  that  is  2f 
feet  wide  be,  to  contain  the  same  number  of  yards  ? 

28.  A  can  mow  an  acre  of  grass  in  4  hours,  B  in  6  hours, 
and  C  in  8  hours.     How  many  days,  working  9  hours  a  day, 
would  they  require  to  mow  39  acres  ? 

ANALYSIS. — Since  A  can  mow  an  acre  in  4  hours,  B  in  6  hours, 
and  C  in  8  hours,  A  can  mow  ^  of  an  acre,  B  ^  of  an  acre,  and 
C  ^  of  an  acre  in  1  hour.  Together  they  can  mow  i-ri+|=H 
of  an  acre  in  1  hour.  And  since  they  can  mow  13  twenty-fourths 
of  an  acre  in  1  hour,  they  can  mow  1  twenty  fourth  of  an  acre 
in  ^  of  1  hour  ;  and  1  acre,  or  f^,  in  24  times  -jV—  ^f,-  of  1  hour  • 
and  to  mow  39  acres,  they  will  require  39  times  ^ — ^  hours, 
which  reduced  to  days  of  9  hours  each,  gives  8  days. 

OPERATION. 

l-H+!=Mhours. 

8    $  n 

v*xyX0  =  8  days.     Or,      $    0 

$  Am.  \  8  days. 

29.  A  can  do  a  piece  of  work  in  4  days,  and  B  can  do  the 
same  in  6  days  ;  in  what  time  can  they  both  do  the  work  if 
they  labor  together  ? 

30.  If  6  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  5  men  to  do  it? 

ANALYSIS. — If  G  men  can  do  a  piece  of  work  in  10  days,  1  man 
will  require  6  times  as  long,  or  60  days  to  do  the  same  work 
Five  men  will  require  but  one  fifth  as  long  as  one  man  or  60^-5 
=-12  days. 


ANALYSIS  207 

OPERATION. 


10x6-^5=12  days. 


6 


Ans.  |  12  days. 


31.  Three  men  together  can  perform  a  piece  of  work  in  9 
days.     A  alone  can  do  it  in  18  days,  B  in  27  days  ;  in  what 
time  can  C  do  it  alone  ? 

32.  A  and  B  can  build  a  wall  on  one  side  of  a  square 
piece  of  ground  in  3  days  ;  A  and  C  in  4  days  ;  B  and  C  in 
6  days :  what  time  will  they  require,  working  together,  to 
complete  the  wall  enclosing  the  square  ? 

33.  Three  men  hire  a  pasture,  for  which  they  pay  66  dol- 
lars.    The  first  puts  in  2  horses  3  weeks  ;  the  second  6  horses 
for  2J  weeks;  the  third  9  horses  for  1J  weeks:  how  much 
ought  eaeh  to  pay  ? 

ANALYSIS. — The  pasturage  of  2  horses  for  3  weeks,  would  he  the 
same  as  the  pasturage  of  1  horse  2  times  3  weeks,  or  6  weeks ; 
that  of  six  horses  2^  weeks,  the  same  as  for  1  horse  6  times  2£ 
weeks,  or  15  weeks  ;  and  that  of  9  horses  1^  weeks,  the  same  as 
1  horse  for  9  times  H  weeks,  or  12  weeks.  The  three  persons  had 
an  equivalent  for  the  pasturage  of  1  horse  for  6+15-f  12 -33  weeks ; 
therefore,  the  first  must  pay  ^j,  the  second  i§,  and  the  third 
41  of  66  dollars 

OPERATION. 

3  x2=6;  then  $66xT£r=$12.  1st 
21x6=15;  "  $66  x  J$ =$30.  2d. 
Ijx9  =  12;  "  $66  x  if =$24.  3d. 

34.  Two  persons,  A  and  B,  cuter  into  partnership,  and  gain 
$175.     A  puts  in  75  dollars  for  4  months,  and  B  puts  in  100 
dollars  for  6  months  :  what  is  each  one's  share  of  the  gain  ? 

35.  Three  men  engage  to  build  a  house  for  580  dollars. 
The  first  one  employed  4  hands,  the  second  5  hands,  and  the 
third  7  hands.     The  first  man's  hands  worked  three  times  as 
many  days  as  the  third,  and  the  second  man's  hands  twice  as 
many  days  as  the  third  man's  hands  :  how  much  must  each 
receive  ? 


208 


ANALYSIS. 


36.  If  8  students  spend  $192  in  6  months,  how  much  will 
12  students  spend  in  20  months  ? 

ANALYSIS. — Since  8  students  spend  $192,  one  student  will  spend 
i  of  $192,  in  6  months ,  in  1  month  1  student  will  spend  -^  of  £ 
of  $192- $4.  Twelve  students  will  spend,  in  1  month,  12  times 
as  much  as  1  student,  and  in  20  months  they  will  spend  20  times 
as  much  as  in  1  month. 


OPERATION. 


24  2 

-w  i  i  n  20 

—  X£XTXyXY=$960. 


48 


20 


$960.  Ans. 


31  If  6  men  can  build  a  wall  80  feet  long,  6  feet  wide, 
and  4  feet  high,  in  15  days,  in  what  time  can  18  men  build 
one  240  feet  long,  8  feet  wide,  and  6  feet  high  ? 

ANALYSIS. — Since  it  takes  6  men  15  days  to  build  a  wall,  it 
will  take  1  man  6  times  15  days,  or  90  days,  to  build  the  same 
wall.  To  build  a  wall  1  foot  long,  will  require  -8\r  as  long  as  to 
build  one  80  feet  long ;  to  build  one  1  foot  wide,  i  as  long  as  to 
build  one  4  feet  wide ;  and  to  build  one  1  foot  high,  £  as  long  as 
to  build  one  6  feet  high,  18  men  can  build  the  same  wall  in  ^ 
of  the  time  that  one  man  can  build  it :  but  to  build  one  240  feet 
long,  will  take  them  240  times  as  long  as  to  build  one  1  foot  in 
length ;  to  build  one  8  feet  wide,  8  limes  as  long  as  to  build  one 
1  foot  wide,  and  to  build  one  C  feet  high,  6  times  as  long  as  to 
build  one  1  foot  high. 


OPERATION. 


$     2 

15x0      1      1     1      1      &<0     $     0  $0 

~~I — X$0X>IX0X;F$  X~T  x;rx  I  ^ 

*  *,! 


15 


Ans.  i  30  days. 

38    If  96/6s.  of  bread  be  sufficient  to  serve  5  men  12  days, 
how  many  days  will  57/6.  serve  19  men? 


ANALYSIS.  209 

39.  If  a  man  travel   220  miles  in  10  days,  travelling  12 
hours  a  day,  in  how  many  days  will  he  travel  880  miles, 
travelling  16  hours  a  day? 

40.  If  a  family  of  12  persons  consume  a  certain  quantity 
of  provisions  in  6  days,  how  long  will  the  same  provisions 
last  a  family  of  8  persons  ? 

41.  If  9  men  pay  $135  for  5  weeks'  board,  how  much 
must  8  men  pay  for  4  weeks'  board  ? 

42.  If  10  bushels  of  wheat  are  equal  to  40  bushels  of 
corn,  and  28  bushels  of  corn  to  56  pounds  of  butter,  and  39 
pounds  of  butter  to  1  cord  of  wood  ;  how  much  wheat  is  12 
cords  of  wood  worth  ? 

ANALYSIS. — Since  10  bushels  of  wheat  are  worth  40  bushels  of 
corn,  1  bushel  of  corn  is  worth  •£>  of  10  bushels  of  wheat,  or 
i  of  a  bushel ;  28  bushels  are  worth  28  times  £  of  a  bushel  of 
wheat,  or  7  bushels :  since  28  bushels  of  corn,  or  7  bushels  of 
wheat  are*worth  56  pounds  of  butter,  1  pound  of  butter  is  worth 
^g  of  7=i  of  a  bushel  of  wheat,  and  39  pounds  are  worth  39 
times  as  much  as  1  pound,  or  39*^=^  bushels  of  wheat;  and 
since  39  pounds  of  butter,  or  ^  bushels  of  wheat  are  worth  1  cord 
of  wood,  12  cords  are  worth  12  times  as  much,  or  12x^=58£ 
bushels. 

OPERATION. 

3 

ro    i    n    i    39  xt 


V  

rf  A  />>      -| 
V  v  i 

2 


39  o 

n  3 


117=5816^. 


NOTE. — Always  commence  analysing  from  the  term  which  is 
of  the  same  name  or  kind  as  the  required  answer. 

43.  If  35  women  can  do  as  much  work  as  20  boys,  and 
16  boys  can  do  as  much  as  7  men  :  how  many  women  can 
do  the  work  of  18  men  ? 

44.  If  36  shillings  in  New  York,  are  equal  to  27  shillings 
in  Massachusetts,  and  24  shillings  in  Massachusetts  are  equal 
to  30  shillings  in  Pennsylvania,  and  45  shillings  in  Pennsyl- 
vania are  equal  to  28  shillings  in  Georgia  ;  how  many  shil- 
lings in  Georgia  are  equal  to  72  shillings  in  New  York  ? 

14 


210  PROMISCUOUS  EXAMPLES 


PROMISCUOUS    EXAMPLES    IN    ANALYSIS. 

1.  How  many  sheep  at  4  dollars  a  head  must  I  give  for  6 
cows,  worth  12  dollars  apiece  ? 

2.  If  7  yards  of  cloth  cost  $49,  what  will  16  yards  cost  ? 

3.  If  36  men  can  build  a  house  in  16  days,  how  long  will 
it  take  12  men  to  build  it? 

4.  If  3  pounds  of  butter  cost  7J  shillings,  what  will  12 
pounds  cost  ? 

5.  If  5 1  bushels  of  potatoes  cost  $2f,  how  much  will  12 J 
bushels  cost  ? 

6.  How  many  barrels  of  apples,  worth  1 2  shillings  a  barrel, 
will  pay  for  16  yards  of  cloth,  worth  9s.  Qd.  a  yard  ? 

7.  If  31 J  gallons  of  molasses  are  worth  $9f ,  what  are  5J 
gallons  worth  ? 

8.  What  is  the  value  of  24|  bushels  of  corn,  at  5s.  *ld.  a 
bushel,  New  York  currency  ? 

9.  How  much  rye,  at  8s.  Zd.  per  bushel,  must  be  given 
for  40  gallons  of  whisky,  worth  2s.  9d.  a  gallon? 

10.  If  it  take  44  yards  of  carpeting,  that  is  1 J  yards  wide, 
to  cover  a  floor,  how  many  yards  of  •£  yards  wide,  will  it 
take  to  cover  the  same  floor  ? 

11.  If  a  piece  of  wall  paper,  14  yards  long  and  1J  feet 
wide,  will  cover  a  certain  piece  of  wall,  how  long  must  an- 
other piece  be,  that  is  2  feet  wide,  to  cover  the  same  wall  ? 

12.  If  5  men  spend  $200  in  160  days,  how  long  will  $300 
last  12  men  at  the  same  rate  ? 

13.  If  1  acre  of  land  cost  £  of  f  of  £  of  $50,  what  will  3| 
acres  cost  ? 

14.  Three  carpenters  can  finish  a  house  in  2  months  ;  two 
of  them  can  do  it  in  2J  months  :  how  long  will  it  take  the 
third  to  do  it  alone  ? 

15.  Three  persons  bought  2  barrels  of  flour  for  15  dollars. 
The  first  one  ate  from  them  2  months,  the  second  3  months, 
and  the  third  7  months  :  how  much  should  each  pay  ? 

16.  What  quantity  of  beer  will  serve  4  persons  18|  days, 
if  6  persons  drink  7£  gallons  in  4  days  ? 


IN  ANALYSIS.  211 

17.  If  9  persons  use  If  pounds  of  tea  in  a  month,  how 
much  will  10  persons  use  in  a  year  ? 

18.  If  |  of  f  of  a  gallon  of  wine  cost  f  of  a  dollar,  what 
will  5  J  gallons  cost  ? 

19.  How  many  yards  of  carpeting,  1|  yards  wide,  will  it 
take  to  cover  a  floor  that  is  4f  yards  wide  and  6  and  three- 
fifths  yards  long  ? 

20.  Three  persons  bought  a  hogshead  of  sugar  containing 
413  pounds.     The  first  paid  $2J  as  often  as  the  second  paid 
$3  J,  and  as  often  as  the  third  paid  $4  :  what  was  each  one's 
share  of  the  sugar  ? 

21.  A,  with  the  assistance  of  B,  can  build  a  wall  2  feet 
wide,  3  feet  high,  and  30  feet  long,  in  4  days  ;  but  with  the 
assistance  of  C,  they  can  do  it  in  3 1  days  :  in  how  many  days 
can  C  do  it  alone  ? 

22.  If  two  persons  engage  in  a  business,  where  one  advances 
$875,  aritt  the  other  $625,  and  they  gain  $300,  what  is  each 
one's  share. 

23.  A  person  purchased  f  of  a  vessel,  and  divided  it  into  5 
equal  shares,  and  sold  each  of  those  shares  for  $1200  :  what 
was  the  value  of  the  whole  vessel  ? 

24.  How  many  yards  of  paper,  f  of  a  yard  wide,  will  be 
sufficient  to  paper  a  room  10  yards  square  and  3  yards  high  ? 

25.  What  will  be  the  cost  of  45#>s.  of  coffee,  New  Jersey 
currency,  if  9?6s.  cost  27  shillings  ? 

26.  What  will  be  the  cost  of  3  barrels  of  sugar,  each  weigh- 
ing %cwt.  at  10c?.  per  pound,  Illinois  currency? 

27.  If  12  men  reap  80  acres  in  6  days,  in  how  many  days 
will  25  men  reap  200  acres  ? 

28.  If  4  men  are  paid  24  dollars  for  3  days'  labor,  how 
many  men  may  be  employed  16  days  for  $96  ? 

29.  If  $25  will  supply  a  family  with  flour  at  $7.50  a  bar- 
rel for  2§  months,  how  long  would  $45  last  the  same  family 
when  flour  is  worth  $6.75  per  barrel  ? 

30.  A  wall  to  be  built  to  the  height  of  27  feet,  was  raised 
to  the  height  of  9  feet  by  1 2  men  in  6  days  :  how  many  men 
must  be  employed  to  finish  the  wall  in  4  days  at  the  same 
rate  of  working  ? 


212  PROMISCUOUS    EXAMPLES. 

31.  A,  B  and  C,  sent  a  drove  of  hogs  to  market,  of  which 
A  owned   105,  B  75,   and  C  120.     On  the  way  60  died  : 
how  many  must  each  lose  ? 

32.  Three  men,  A,  B  and  C,  agree  to  do  a  piece  of  work, 
for  which  they  are  to  receive  $315.     A  works  8  days,  10  J 
hours  a  day  ;  B  9  j  days,  8  hours  a  day  ;  and  C,  4  days,  12 
hours  a  day  :  what  is  each  one's  share  ? 

33.  If  1 0  barrels  of  apples  will  pay  for  5  cords  of  wood, 
and  12  cords  of  wood  for  4  tons  of  hay,  how  many  barrels  of 
apples  will  pay  for  9  tons  of  hay  ? 

34.  Out  of  a  cistern  that  is  f  full  is  drawn  140  gallons, 
when  it  is  found  to  be  \  full :  how  much  does  it  hold  ? 

35.  If  .7  of  a  gallon  of  wine  cost  $2.25,  what  will  .25  of  a 
gallon  cost  ? 

36.  If  it  take  5.1  yards  of  cloth,  1.25  yards  wide,  to  make  a 
gentleman's  cloak,   how  much  surge,  f  yards  wide,  will  be 
required  to  line  it  ? 

37.  A  and  B  have  the  same  income.     A  saves  |  of  his 
annually  ;  but  B,  by  spending  $200  a  year  more  than  A,  at 
the  end  of  5  years  find  himself  $160  in  debt :  what  is  their 
income  ? 

38.  A  father  gave  his  younger  son  $420,  which  was  |  of 
what  he  gave  to  his  elder  s.on  ;  and  3  times  the  elder  son's 
portion  was  \  the  value  of  the  father's  estate  :  what  was  the 
value  of  the  estate  ? 

39.  Divide  $176.40  among  3  persons,  so  that  the  first  shall 
have  twice  as  much  as  the  second,  and  the  third  three  times 
as  much  as  the  first :  what  is  each  one's  share  ? 

40.  A  gentleman  having  a  purse  of  money,  gave  \  of  it  for 
a  span  of  horses  ;  £  of  £  of  the  remainder  for  a  carriage  : 
when  he  found  that  he  had  but  $100  left :  how  much  was  in 
his  purse  before  any  was  taken  out  ? 

41.  A  merchant  tailor  bought  a  number  of  pieces  of  cloth, 
each  containing  25^  yards,  at  the  rate  of  3  yards  for  4  dol- 
lars, and  sold  them  at  the  rate  of  5  yards  for  13  dollars,  and 
gained  by  the  operation  96  dollars  :  how  many  pieces  did  he 
buy? 


RATIO   AND   PROPORTION.  213 


RATIO    AND    PROPORTION. 

221.  Two  numbers  having  the  same  unit,  may  be  com- 
pared in  two  ways : 

1st.  By  considering  how  much  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference  ;  and, 

3d.  By  considering  how  many  times  one  is  contained  in  the 
other,  which  is  shown  by  their  quotient. 

In  comparing  two  numbers,  one  with  the  other,  by  means 
of  their  difference,  the  less  is  always  taken  from  the  greater. 

In  comparing  two  numbers,  one  with  the  other,  by  means 
of  their  quotient,  one  of  them  must  be  regarded  as  a  standard 
which  measures  the  other,  and  the  quotient  which  arises  by 
dividing  by  the  standard,  is  called  the  ratio. 

222.  Every  ratio  is  derived  from  two  numbers :  the  first 
is  called  the  antecedent,  and  the  second  the  consequent:  each 
is  called  a  term,  and  the  two,  taken  together,  are  called  a 
couplet.     The  antecedent  will  be  regarded  as  the  standard. 

If  the  numbers  3  and  12  be  compared  by  their  difference, 
the  result  of  the  comparison  will  be  9  ;  for,  12  exceeds  3  by  9. 
If  they  are  compared  by  means  of  their  quotient,  the  result 
will  be  4  ;  for,  3  is  contained  in  12,  4  tunes :  that  is, 
3  measuring  12,  gives  4. 

223.  The  ratio  of  one  number  to  another  is  expressed  in 
two  ways : 

1st.  By  a  colon  ;  thus,  3  :  12  ;  and  is  read,  3  is  to  12  ;  or, 

o  measuring  12. 

12 

2d.  In  a  fractional  form,  as—;  or,  3  measuring  12. 


231.  In  how  many  ways  may  two  numbers,  having  the  same  unit,  be 
compared  with  each  other  ?  If  you  compare  by  their  difference,  how  do 
you  find  it  ?  If  you  compare  by  the  quotient,  how  do  you  regard  one  of 
the  numbers  ?  What  is  the  ratio  ? 

222.  From  how  many  terms  is  a  ratio  derived  ?  What  is  the  first 
term  called  ?  What  is  the  second  called  ?  Which  is  the  standard  ? 

2~53.  How  may  the  ratio  of  two  numbers  be  expressed  ?    How  read  ? 


214  RATIO   AND   PROPORTION. 

224.  If  two  couplets  have  the  same  ratio,  their  terms  are 
said  to  be  proportional :  the  couplets 

3     :     12    and    1     :     4 

have  the  same  ratio  4  ;  hence,  the  terms  arc  proportional, 
and  are  written, 

3     :     12     :     :     1     :     4 

by  simply  placing  a  double  colon  between  the  couplets.     The 
terms  are  read 

3  is  to  12     as     1  is  to  4, 
and  taken  together,  they  are  called  a  proportion :  hence, 

A  proportion  is  a  comparison  of  the  terms  of  two  equal 
ratios* 

224.  If  two  couplets  have  the  same  ratio,  what  is  said  of  the  terms  ? 
How  are  they  written  V  How  read  ?  What  is  a  proportion  ? 

*  Some  authors,  of  high  authority,  make  the  consequent  the  stand- 
ard and  divide  the  antecedent  by  it  to  determine  the  ratio  of  the  couplet. 

The  ratio  3  :  13  is  the  same  as  that  of  1:4  by  both  methods ; 
for,  if  the  antecedent  be  made  the  standard,  the  ratio  is  4 ;  if  the  conse- 
quent be  made  the  standard,  the  ratio  is  one-fourth.  The  question  is, 
which  method  should  be  adopted  V 

The  unit  1  is  the  number  from  which  all  other  numbers  are  derived, 
and  by  which  they  are  measured. 

The  question  is,  how  do  we  most  readily  apprehend  and  express  the 
relation  between  1  and  4  ?  Ask  a  child,  and  he  will  answer,  "the  dif- 
ference is  3."  But  when  you  ask  him,  "how  many  1's  are  there  in 
4V"  he  will  answer,  "4,"  using  1  as  the  standard. 

Thus,  we  begin  to  teach  by  using  the  standard  1 :  that  is,  by  dividing 
4byl. 

Now,  the  relation  between  3  and  13  is  the  same  as  that  between  1 
and  4;  if  then,  we  divide  4  by  1,  we  must  also  divide  13  by  3.  Do  we, 
indeed,  clearly  apprehend  the  ratio  of  3  to  12,  until  we  have  referred  to 
1  as  a  standard  ?  Is  the  mind  satisfied  until  it  has  clearly  perceived  that 
the  ratio  of  3  to  13  is  the  same  as  that  of  1  to  4  ? 

In  the  Rule  of  Three  we  always  look  for  the  result  in  the  4th  term. 
Now,  if  we  wish  to  find  the  ratio  of  3  to  13,  by  referring  to  1  as  a  stand- 
ard, we  have 

3    :    13    :    :    1    :    ratio, 

which  brings  the  result  in  the  right  place. 

But  if  we  define  ratio  to  be  the  antecedent  divided  by  the  consequent, 
we  should  have 

3    :    12    :    :    ratio    :    1, 

which  would  bring  the  ratio,  or  required  number,  in  the  3d  place, 


RATIO   AND   PROPORTION.  215 

What  are  the  ratios  of  the  proportions, 

3  :  9  :  :  12  :  36? 
2  :  10  :  :  12  :  60? 

4  :  2  :  :  8  :  4? 
9  :  1  :  <  90  :  10? 

225.  The  1st  and  4th  ter-ms  of  a  proportion  are  called  the 
extremes  :  the  2d  and  3d  terms,  the  means.  Thus,  in  the  pro- 
portion, 

3     :     12     :     :     6     :     24 

3  and  24  are  the  extremes,  and  12  and  6  the  means: 

12     24 

Since  (Art.  224),  Y^lp 

we  shall  have,  by  reducing  to  a  common  denominator, 
12x6_24x3 
!Tx~6~  6x3' 

But  since  the  fractions  are  equal,  and  have  the  same  deno- 
minators, their  numerators  must  be  equal,  viz.  ; 

12x6=24x3;  that  is, 

In  any  proportion,  the  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

Thus,  in  the  proportions, 

1   :     6  :   :    2  :  12  ;  we  have  1  x  12=  6x2; 
4   :  12   :   :    S   :  24  ;     "     "     4x24  =  12x8. 

220.  Since,  in  any  proportion,  the  product  of  the  extremes 
is  equal  to  the  product  of  the  means,  it  follows  that, 

In  all  cases,  the  numerical  value  of  a  quantity  is  the  number  of  times 
which  that  quantity  contains  an  assumed  standard,  called  its  unit  of 


If  we  would  find  that  numerical  value,  in  its  right  place,  we  must 
say, 

standard  :  quantity  :  :  1  :  numerical  value  : 
but  if  we  take  the  other  method,  we  have 

quantity  :  standard  :  :  numerical  value  :  1, 
which  brings  the  numerical  value  in  the  wrong  place. 


216  RATIO   AND   PROPORTION". 

1st.  If  the  product  of  the  means  be  divided  by  one  of  the 
extremes,  the  quotient  will  be  the  other  extreme. 

Thus,  in  the  proportion 

3  :  12   :   :  6:  24,  we  have  3  x  24  =  12  x  6  ; 

then,  if  12,  the  product  of  the  means,  be  divided  by  one  of 
the  extremes,  3,  the  quotient  will  be  the  other  extreme,  24  : 
or,  if  the  product  be  divided  by  24,  the  quotient  will  be  3. 

2d.  If  the  product  of  the  extreme?  be  divided  by  either  of 
the  means,  the  quotient  ivill  be  the  other  mean. 

Thus,  if  3  x  23=12  x  6  =  72  be  divided  by  12,  the  quotient 
will  be  6  or  if  it  be  divided  by  6,  the  quotient  will  be  12. 

EXAMPLES. 

1.  The  first  three  terms  of  a  proportion  are  3,  9  and  12  : 
what  is  the  fourth  term  ? 

2  The  first  three  terms  of  a  proportion  are  4,  16  and  15  : 
what  is  the  4th  term  ? 

3.  The  first,  second,  and  fourth  terms  of  a  proportion  are 
6,  12  and  24  :  what  is  the  third  term  ? 

4.  The  second,  third,  and  fourth  terms  of  a  proportion  are 
9,  6  and  24  :  what  is  the  first  term  ? 

5.  The  first,  second  and  fourth  terms  are  9,  18  and  48  : 
what  is  the  third  term  ? 

227.  Simple  and  Compound  Eatio. 

The  ratio  of  two  single  numbers  is  called  a  Simple  Eatio, 
.and  the  proportion  which  arises  from  the  equality  of  two  such 
ratios,  a  Simple  Proportion. 


225.  Which  are  the  extremes  of  a  proportion  ?    Which  the  means  ? 
What  is  the  product  of  the  extremes  equal  to  ? 

226.  If  the  product  of  the  means  be  divided  hy  one  of  the  extremes, 
what  will  the  quotient  be  ?    If  the  product  of  the  means  be  divided  by 
either  extreme,  what  will  the  quotient  be  ? 

227.  What  is  a  simple  ratio  ?    What  is  the  proportion  called  which 
•comes  from  the  equality  of  two  simple  ratios?    What  is  a  compound 

ratio  ?    What  is  a  compound  proportion  ? 


RATIO   AND   PROPORTION.  217 

If  the  terms  of  one  ratio  be  multiplied  by  the  terms  of  an- 
other, antecedent  by  antecedent  and  consequent  by  conse- 
quent, the  ratio  of  the  products  is  called  a  Compound  Ratio- 
Thus,^if  the  two  ratios 

3     :     6  and  4     :     12 

be  multiplied  together,  we  shall  have  the  compound  ratio 
3x4     :     6x12,  or  12     :     72  ; 

In  which  the  ratio  is  equal  to  the  product  of  the  simple 
ratios. 

A  proportion  formed  from  the  equality  of  two  compound 
ratios,  or  from  the  equality  of  a  compound  ratio  and  a  simple 
ratio,  is  called  a  Compound  Proportion. 

228.   What  part  one  number  is  of  another. 

When  the  standard,  or  antecedent,  is  greater  than  the 
number  which  it  measures,  the  ratio  is  a  proper  fraction, 
and  is  such  a  part  of  1,  as  the  number  measured  is  of  the 
standard. 

1.  What  part  of  12  is  3  ?  that  is,  what  part  of  the  stand- 
ard 12,  is  3  ? 


12     :     3      :      :     1      :     I; 
that  is,  the  number  measured  is  one-fourth  of  the  standard. 


2.  What  part  of  9  is  2  ? 

3.  What  part  of  16  is  4? 

4.  What  part  of  100  is  20  ? 

5.  What  part  of  300  is  200  ? 

6.  What  part  of  36  is  144  ? 


7.  3  is  what  part  of  12  ? 


8.  5  is  what  part  of  20  ? 

9.  8  is  what  part  of  56  ? 

10.  9  is  what  part  of  8  ? 

11.  12  is  what  part  of  132  ? 


NOTE. — The  standard  is  generally  preceded  by  the  word  of,  and 
in  comparing  numbers,  may  be  named  second,  as  in  examples  7, 
8,  1),  10  and  11,  but  it  must  be  always  be  used  as  a  divisor,  and 
should  be  placed  first  in  the  statement. 


238.  When  the  standard  is  greater  than  the  consequent,  how  may 
the  ratio  be  compared  ?  What  part  is  3  of  1  ?  5  of  1  ?  What  part  is 
4  of  2  ?  12  of  3  ?  7  of  5  ? 


218 


SINGLE   RULE   OF   THREE. 


SINGLE  RULE  OF  THREE. 

229.  The  Single  Rule  of  Three  is  an  application  of  the 
principle  of  simple  ratios.  Three  numbers  are  always  given 
aixl  a  fourth  required.  The  ratio  between  two  of  the  given 
numbers  is  the  same  as  that  between  the  third  and  the  required 
number. 


1.  If  3  yards  of  cloth  cost  $12,  what  will  6  yards  cost  at  the 
same  rate  ? 

NOTE. — We  shall  denote  the  required  term  of  tlie  proportion  by 
the  letter  x. 


STATEMENT. 

yd.  yd.         $ 
3   :  6   :  :   12 

OPERATION. 
12  o 
0 


:    x 


ANALYSIS. — The  condition,  "  at  the  same 
rate,"  requires  that  the  quantity  3  yards 
must  have  the  same  ratio  to  the  quantity  6 
yards,  as  $12,  the  cost  of  3  yards,  to  x  dol- 
lars, the  cost  of  12  yards. 

Since  the  product  of  the  two  extremes  is 
equal  to  the  product  of  the  two  means,  (Art. 
235),   3xz=Gxl2;    and  if   3x^=6x12,   x 
must  be  equal  to  this  product  divided  by  3 :      A^C      J-AQA 
that  is, 

The  4th  term  is  equal  to  the  product  of  the  second  and  third 
terms  divided  by  the  first. 

2.  If  56  dollars  will  buy  14  yards  of  broadcloth,  how  many 
yards,  at  the  same  rate,  can  be  bought  for  84  dollars  ? 


ANALYSIS.— Fifty-six  dollars,  (being 
the  cost  of  14  yards  of  cloth),  has  the 
same  ratio  to  $84,  as  14  yards  has  to  the 
number  of  yards  which  $84  will  buy 

NOTE. — When  the  vertical  line  is  used, 
the  required  term,  (which  is  denoted  by 
a;),  is  written  on  the  left 


STATEMENT. 

$         $          yd.  yd. 
56   :  84   :   :   14   :   x 

OPERATION 


21 


229.  What  is  the  Single  Rule  of  Three  ?  How  many  numbers  are 
§fivcn  ?  How  many  required  ?  What  ratio  exists  between  two  of  the 
given  numbers  ? 


SINGLE    RULE   OF    THREE.  219 

230.  Hence,  we  have  the  following 

RULE  I.  Write  the  number  which  is  of  the  same  kind  with 
the  answer  for  the  third  term,  the  number  named  in  connection 
with  it  for  the  first  term,  and  the  remaining  number  for  the 
second  term. 

II.  Multiply  the  second  and  third  terms  together,  and  divide 
the  product  by  the  first  term :  Or, 

Multiply  the  third  term  by  the  ratio  of  the  first  and  second. 

NOTES. — 1.  If  the  first  and  second  terms  have  different  units, 
they  must  be  reduced  to  the  same  unit. 

2.  If  the  third  term  is  a  compound  denominate  number,  it  must 
be  reduced  to  its  smallest  unit. 

3.  The  preparation  of  the  terms,  and  writing  them  in  their  pro- 
per places,  is  called  the  statement. 

EXAMPLES. 

1.  If  I  can  walk  84  miles  in  3  days,  how  far  can  I  walk  in 
11  days? 

2.  If  4  hats  cost  $12,  what  will  be  the  cost  of  55  hats  at 
the  same  rate  ? 

3.  If  40  yards  of  cloth  cost  $170,  what  will  325  yards  cost 
at  the  same  rate  ? 

4.  If  240  sheep  produce  660  pounds  of  wool,  how  many 
pounds  will  be  obtained  from  1200  sheep? 

5    If  2  gallons  of  molasses  cost  65  cents,  what  will  3  hogs- 
heads cost  ? 

6.  If  a  man  travels  at  the  rate  of  210  miles  in  6  days,  how 
far  will  he  travel  in  a  year,  supposing  him  not  to  travel  on 
Sundays  ? 

7.  If  4  yards  of  cloth  cost  $13,  what  will  be  the  cost  of  3 
pieces,  each  containing  25  yards  ? 

8.  If  48  yards  of  cloth  cost  $67.25,  what  will  144  yards 
cost  at  the  same  rate  ? 

9.  If  3  common  steps,  or  paces,  are  equal  to  2  yards,  how 
many  yards  are  there  in  1 60  paces  ? 

10.  If  750  men  require  22500  rations  of  bread  for  a  month, 
how  many  rations  will  a  garrison  of  1200  men  require  ? 

235.  Give  the  rule  for  the  statement.    Give  the  rule  for  finding  the 
fourth  term. 


220  SINGLE    RULE   OF   THREE. 

11.  A  cistern  containing  200  gallons  is  filled  by  a  pipe 
which  discharges  3  gallons  in  5  minutes  ;  but  the  cistern  has 
a  leak  which  empties  at  the  rate  of  1  gallon  in  5  minutes. 
If  the  water  begins  to  run  in  when  the  cistern  is  empty,  how 
long  will  it  run  before  filling  the  cistern  ? 

12.  If  14|  yards  of  cloth  cost  $19*,  how  much  will  19  J 
yards  cost  ? 

NOTE. — First  make  the  STATEMENT. 

statement ;  then  change  tlio  yd.        yd,  $          $ 

mixed     numbers     to    im-  \±\    :  \C)1    .    :    IQi    :   % 

proper  fractions,  after 
which  arrange  the  terms, 
and  cancel  equal  factors 
according  to  previous  in-  „ 

struction. 


13.  If  •§-  of  a  yard  of  cloth  cost  -£  of  a  dollar,  what  will 
2 \  yards  cost? 

14.  If  y\  of  a  ship  cost  £273  2s.  Qd.,  what  will  ^  of  her 
cost  ? 

15.  If  1T4T  bushels  of  wheat  cost  $2*,  how  much  will  60 
bushels  cost  ? 

16.  If  4|  yards  of  cloth  cost  $9.15,  what  will  13|  yards 
cost? 

17.  If  a  post  8  feet  high  cast  a  shadow  12  feet  in  length, 
what  must  be  the  height  of  a  tree  that  casts  a  shadow  122 
feet  in  length,  at  the  same  time  of  day  ? 

18.  If  ^cwt.  Iqr.  of  sugar  cost  $64.96,  what  will  be  the 
cost  of  kcwt.  2qr.  ? 

19.  A  merchant  failing  in  trade,  pays  65  cents  for  every 
dollar  which  he  owes  :    he  owes  A  $2750,  and  B  $1975  : 
how  much  does  he  pay  each  ? 

20.  If  6  sheep  cost  $15,  and  a  lamb  costs  one-third  as 
much  as  a  sheep,  what  will  27  lambs  cost? 

21.  If  2/6s.  of  beef  cost  J  of  a  dollar,  what  will   30/6*. 
cost? 

22.  If  4-J-  gallons  of  molasses  cost  $2f ,  how  much  is  it  per 
quart  ? 

23.  A  man  receives  f  of  his  income,  and  finds  it  equal  to 
$3724.16  :  how  much  is  his  whole  income  ? 


SINGLE    RULE   OF   THREE.  221 

24.  If  4   barrels   of  flour  cost  $34 f,  how  much   can  be 
bought  for  $175£? 

25.  If  2  gallons  of  molasses  cost  65  cents,  what  will  3 
hogsheads  cost  ? 

26.  What  is  the  cost  of  -6  bushels  of  coal  at  the  rate  of 
£1  Us.  Qd.  a  chaldron? 

27.  What  quantity  of  corn  can  I  buy  for  90  guineas,  at  the 
rate  of  6  shillings  a  bushel  ? 

28.  A  merchant   failing  in   trade    owes  $3500,  and  his 
effects  are   sold  for  $2100  :  how  much  does  B.  receive,  to 
whom  he  owes  $420  ? 

29.  If  3  yards  of  broadcloth  cost  as  much  as  4  yards  of 
cassimere,  how  much  cassimere  can  be  bought  for  18  yards 
of  broadcloth  ? 

30.  If  7  hats  cost  as  much  as  25  pair  of  gloves,  worth  84 
cents  a  pair,  how  many  hats  can  be  purchased  for  $216  ? 

31.  How  many  barrels  of  apples  can  be  bought  for  $114.33, 
if  7  barrels  cost  $21.63? 

32.  If  27  pounds  of  butter  will  buy  45  pounds  of  sugar, 
how  much  butter  will  buy  36  pounds  of  sugar  ? 

33.  If  42J  tons  of  coal  cost  $206.21,  what  will  be  the  cost 
of  2J  tons  ? 

34.  If  40  gallons  run  into  a  cistern,  holding  700  gallons,  in 
an  hour,  and  15  run  out,  in  what  time  will  it  be  filled  ? 

35.  A  piece  of  land  of  a  certain  length  and  12 J  rods  in 
width,  contains  1 J  acres,  how  much  would  there  be  in  a  piece 
of  the  same  length  26 f  rods  wide  ? 

36.  If  13  men  can  be  boarded  1  week  for  $39,585,  what 
will  it  cost  to  board  3  men  and  6  women  the  same  time,  the 
women  being  boarded  at  half  price  ? 

37.  What  will  75  bushels  of  wheat  cost,  if  4  bushels  3 
pecks  cost  $10.687? 

38.  What  will  be  the  cost,  in  United  States  money,  of  324 
yards  3qrs.  of  cloth,  at  5s.  ±d.  New  York  currency,  for  2 
yards  ? 

39.  At  $1.12J  a  square  foot,  what  will  it  cost  to  pave  a 
floor  18  feet  long  and  12ft.  (tin.  wide  ? 


222  CAUSE   AND   EFFECT. 


CAUSE  AND  EFFECT. 

231.  Whatever  produces  effects,  as  men  at  work,  animals 
eating,  time,  goods  purchased  or  sold,  money  lent,  and  the 
like,  may  be  regarded  as  causes. 

Causes  are  of  two  kinds,  simple  and  compound. 

A  simple  cause  has  but  a  single  element,  as  men  at  work,  a 
portion  of  time,  goods  purchased  or  sold,  and  the  like. 

A  compound  cause  is  made  up  of  two  or  more  simple  ele- 
ments, such  as  men  at  work  taken  in  connection  with  time,  and 
the  like. 

232.  The  results  of  causes,  as  work  done,  provisions  con- 
sumed, money  paid,  cost  of  goods,  and  the  like,  may  be  re- 
garded as  effects.     A  simple  effect  is  one  which  has  but  a 
single  element ;  a  compound  effect  is  one  which  arises  from 
the  multiplication  of  two  or  more  elements. 

233.  Causes  which  are  of  the  same  kind,  that  is,  which  can 
be  reduced  to  the  same  unit,  may  be   compared  with  each 
other  ;  and  effects  which  are  of  the  same  kind  may  likewise 
be  compared  with  each  other.     From  the  nature  of  causes  and 
effects,  we  know  that 

1st  Cause  :  2d  Cause  :    :  1st  Effect  :  2d  Effect ; 
and,  1st  Effect  :  2d  Effect  :    :  1st  Cause  :  2d  Cause. 

234.  Simple  causes  and  simple  effects  give  rise  to  simple 
ratios.     Compound  causes  or  compound  effects  give  rise  to 
compound  ratios. 


331.  What  arc    causes?     How  many  kinds  of    causes  are  there? 
What  is  a  simple  cause  ?    What  is  a  compound  cause  ? 
1    232.  What  are  effects?    What  is  a  simple  effect?    What  is  a  com- 
pound effect? 

233.  What  causes  are  of  the  same  kind  ?    What  causes  may  be  com- 
pared with  each  other  ?    What  do  we  infer  from  the  nature  of  causes 
and  effects  ? 

234.  What  gives  rise  to  simple  ratios  ? 


DOUBLE  RULE  OF  THREE.         223 

DOUBLE  RULE  OF  THREE. 

236.  The  Double  Rule  of  Three  is  an  application  of  the 
principles  of  compound  proportion.     It  embraces  all  that  class 
of  questions  in  which  the  causes  are  compound,  or  in  which 
the  effects  are  compound  ;  arid  is  divided  into  two  parts  : 

1st    When  the  compound  causes  produce  the  same  effects  ; 
2<2.  When  the  compound  causes  produce  different  effects. 

237.  When  the  compound  causes  produce  the  same  effects. 
1.  If  6  men  can  dig  a  ditch  in  40  days,  what  time  will  30 

men  require  to  dig  the  same  ? 


ANALYSIS. — The  first  cause 


STATEMENT. 


men.     men. 


is  compounded  of  6  men,  and  «    ' .      on 

•40  days,  the  time  required  to  °      :     OIJ 

do  the  work,  and  n  equal  to  days.   days. 

what    1    man   would    do    in  40      :      x 

G  x  40=240  days.  240     :    30  xx 

The  second  cause  is  com- 
pounded of  30  men  and  the 
number  of  days  necessary  to  #0 

do    th'}     same    work,     viz :  x 


ditch,     ditch. 
:  1     :     i 


But  since  the  effects  are  the  x~  8  davs- 

same,  viz :  the  work  done,  the  causes  must  be  equal ;  hence,  the 
products  of  the  elements  of  the  causes  are  equal.  Therefore,  in  the 
solution  of  all  like  examples, 

Write  the  cause  containing  the  unknown  element  on  the  left 
of  the  vertical  line  for  a  divisor,  and  the  other  cause  on  the 
right  for  a  dividend. 

NOTE. — This  class  of  questions  has  generally  been  arranged 
under  the  head  of  "  Rule  of  Three  Inverse." 

EXAMPLES. 

1.  A  certain  work  can  be  done  in  12  days,  by  working  4 
hours  a  day  :  how  many  days  would  it  require  the  same 
number  of  men  to  do  the  same  work,  if  they  worked  6  hours 
a  day? 

336.  What  is  the  double  Rule  of  Three  ?    What  class  of  questions 
does  it  embrace  ?    Into  how  many  parts  is  it  divided  ?    What  are  they  ? 

337.  What  is  the  rule  when  the  effects  are  equal  ?    Under  what  rule 
has  this  class  of  cases  been  arranged  ? 


224:  DOUBLE   RULE   OF   THREE. 

2.  A  pasture  of  a  certain  extent  supplies  30  horses  for  18 
days :  how  long  will  the  same  pasture  supply  20  horses  ? 

3.  If  a  certain  quantity  of  food  will  subsist  a  family  of  12 
persons  48  days,  how  long  will  the  same  food  subsist  a  family 
of  8  persons  ? 

4.  If  30  barrels  of  flour  will  subsist  100  men  for  40  days, 
how  long  will  it  subsist  25  men  ? 

5.  If  90  bushels  of  oats  will  feed  40  horses  for  six  days, 
how  many  horses  would  consume  the  same  in  1 2  days  ? 

6.  If  a  man  perform  a  journey  of  22  J  days,  when  the  days 
are  12  hours  long,  how  many  days  will  it  take  him  to  per- 
form the  same  journey  when  the  days  are  15  hours  long? 

7.  If  a  person  drinks  20  bottles  of  wine  per  month  when  it 
costs  2s.  per  bottle,  how  much  must  he  drink  without  increas-  • 
ing  the  expense  when  it  costs  2s.  6e?.  per  bottle  ? 

8.  If  9  men  in  18  days  will  cut  150  acres  of  grass,  how 
many  men  will  cut  the  same  in  27  days  ? 

9.  If  a  garrison  of  536  men  have  provisions  for  326  days, 
how  long  will  those  provisions  last  if  the  garrison  be  increased 
to  1304  men  ? 

10.  A  pasture  of  a  certain  extent  having  supplied  a  body 
of  horse,  consisting  of  3000,  with  forage  for  18  days  :  how 
many  days  would  the  same  pasture  have  supplied  a  body  of 
2000  horse  ? 

11.  What  length  must  be  cut  off  from  a  board  that  is  9 
inches  wide,  to  make  a  square  foot,  that  is,  as  much  as  is 
contained  in  12  inches  in  length  and  12  in  breadth  ? 

12.  If  a  certain  sum  of  money  will  buy  40  bushels  of  oats 
at  45  cents  a  bushel,  how  many  bushels  of  barley  will  the 
same  money  buy  at  72  cents  a  bushel  ? 

13.  If  30  barrels  of  flour  will  support  100  men  for  40 
days,  how  long  would  it  subsist  400  men  ? 

14.  The  governor  of  a  besieged  place  has  provisions  for  54 
days,  at  the  rate  of  2/6.  of  bread  per  day,  but  is  desirous  of 
prolonging  the  siege  to  80  days  in  expectation  of  succor :  what 
must  be  the  ration  of  bread  ? 


DOUBLE  RULE   OF  THREE. 


225 


238.  When  the  Compound  Causes  produce  different 
Effects. 

In  this  class  of  questions,  either  a  cause,  or  a  single  ele- 
ment of  a  cause  may.  be  required  ;  or  an  effect,  or  a  single 
element  of  an  effect  may  be  required. 

1.  If  a  family  of  6  persons  expend  $300  in  8  months,  how 
much  will  serve  a  family  of  15  persons  for  20  months  ? 


ANALYSIS. — In  this  example  the  second 
effect  is  required  ;  and  the  statement  may  be 
read  thus  :  If  6  persons  in  8  months  expend 
$300,  15  persons  in  20  months  will  expend 
how  many  (or  x)  dollars  ? 


OPERATION 

15     5 

(  *0     &  25 

X 


#=1875  Ans. 


STATEMENT. 

1st  Cause  :  2d  Cause  :  :  1st  Effect  :  2d  Effect 


15) 

20  j 


Or,  6x8     :.    15x20 


$300 
300 


2.  If  16  men,  in  12  days,  build  18  feet  of  wall,  how  many 
men  must  be  employed  to  build  72  feet  in  8  days  ? 

ANALYSIS. — In  this  example  an  element  of 
the  second  cause  is  required,  viz  :  the  number 
of  men.  The  question  may  be  read  thus : 
If  16  men,  in  12  days,  build  18  feet  of  wall, 
how  many  (or  x)  men,  in  8  days,  will  build 
72  feet  of  wall  ? 


.  , 

*  $ 
$ 


x 


OPERATION. 
^  ,4 
"  *  9 
Jf 
12 


=96  men. 


STATEMENT. 

1}     ••••  1S  ••  ^ 

1  Q  •  ^79 

.      io  .  \  A. 


in     , 

12  j 
Or,  16  x  12         : 

3.  If  32  men  build  a  wall  36  feet  long,  8  feet  high,  and 
4  feet  thick,  in  4  days,  working  12  hours  a  day  •  how  long 
a  wall,  that  is  6  feet  high,  and  3  feet  thick  can  48  men  build 
in  36  days,  working  9  hours  a  day  ? 


238.     When  the  compound  causes  produce  different  effects,  what  will 
always  be  required  ? 
15 


226  DOUBLE   BULE   OF  THKEE.   ' 

OPERATION. 


)  48  36)  x 

Y      :     36     :   :       8>      :     6 
)  9'  4)  3 


ANALYSIS. — In  this  example  an  element  of  the 
second  effect  is  required,  viz  :  the  length  of  the 
wall,  and  the  question  may  be  read  thus  :  If 
32  men,  in  4  days,  working  12  hours  a  day, 
can  build  a  wall  36  feet  long,  8  feet  high,  and 
4  feet  thick,  48  men  in  36  days,  working  9 
hours  a  day,  can  build  a  wall  how  many  (or  x) 
feet  long,  6  feet  high,  and  3  feet  thick  ? 

#1=648  feet. 

STATEMENT. 

32 
4 
12 

Or,  32x4x12    :   48x36x9   :   :   36x8x4    :  #x6x3. 
239.  Hence,  we  have  the  following 

RULE. — I.  Arrange  the  terms  in  the  statement  so  that  the 
causes  shall  compose  one  couplet,  and  the  effects  the  other, 
putting  x  in  the  place  of  the  required  element  : 

II.  Then  if  x  fall  in  one  of  the  extremes,  make  the 
product  of  the  means  a  dividend,  and  the  product  of  the 
extremes  a  divisor;  but  if  x  fall  in  one  of  the  means,  make 
the  product  of  the  extremes  a  dividend,  and  the  product  of 
the  means  a  divisor. 

EXAMPLES. 

1.  If  I  pay  $24  for  the  transportation  of  96  barrels  of  flour 
200  miles,  what  must  I  pay  for  the  transportation  of  480  bar- 
rels 75  miles  ? 

2.  If  12  ounces  of  wool  be  sufficient  to  make  1|  yards  of 
cloth  6  quarters  wide,  what  number  of  pounds  will  be  required 
to  make  450  yards  of  flannel  4  quarters  wide  ? 

3.  What  will  be  the  wages  of  9  men  for  11  days,  if  the 
wages  of  6  men  for  14  days  be  $84  ? 

4.  How  long  would  406  bushels  of  oats  last  7  horses,  if  154 
bushels  serve  14  horses  44  days  ? 

£.  If  a  man  travel  217  miles  in  7  days,  travelling  6  hours 
7  tfay,  how  far  would  he  travel  in  9  days  if  he  travelled  11 
fiours  a  day  ? 

939.  What  is  the  rule  for  finding  tho  unknown  part  ? 


DOUBLE  SULE  OF  THREE.          227 

6.  If  27  men  can  mow  20  acres  of  grass  in  5$-  days,  work- 
ing 3f  hours  a  day,  how  many  acres  can  10  men  mow  in  4| 
days,  by  working  8  J  hours  a  day  ? 

7.  How  long  will  it  take  5  men  to  earn  $11250,  if  25  men 
can  earn  $6250  in  2  years  ? 

8.  If  15  weavers,  by  working  10  hours  a  day  for  10  days, 
can  make  250  yards  of  cloth,  how  many  must  work  9  hours 
a  day  for  15  days  to  make  60 7 J  yards? 

9.  A  regiment  of  100  men  drank  20  dollars'  worth  of  wine 
at  30  cents  a  bottle  :  how  many  men,  drinking  at  the  same 
rate,  will  require  1 2  dollars'  worth  at  25  cents  a  bottle  ? 

10.  If  a  footman  travel  341  miles  in  7^  days,  travelling 
12  J  hours  each  day,  in  how  many  days,  travelling  10^  hours 
a  day,  will  he  travel  155  miles? 

11.  If  25  persons  consume  300  bushels  of  corn  in  1  year, 
how  much  will   139  persons  consume  in  8  months,  at  the 
same  rate  ? 

12.  How  much  hay  will  32  horses  eat  in  120  days,  if  96 
horses  eat  3J  tons  in  7|  weeks  ? 

13.  If  $2. 45  will  pay  for  painting  a  surface  21  feet  long 
and  13 J  feet  wide,  what  length  of  surface  that  is  lOf  feet 
wide,  can  be  painted  for  $31.72  ? 

14.  How  many  pounds  of  thread  will  it  require  to  make 
60  yards  of  3  quarters  wide,  if  7  pounds  make  14  yards 
6  quarters  wide  ? 

15.  If  500  copies  of  a  book,  containing  210  pages,  require 
12  reams  of  paper,  how  much  paper  will  be  required  to  print 
1200  copies  of  a  book  of  280  pages? 

16.  If  a  cistern  17J  feet  long,  10£  feet  wide,  and  13  feet 
deep,  hold  546  barrels  of  water,  how  many  barrels  will  a 
cistern  12  feet  long,  10  feet  wide,  and  7  feet  deep,  contain  ? 

17.  A  contractor  agreed  to  build  24  miles  of  railroad  in  8 
months,  and  for  this  purpose  employed  150  men.     At  the 
end  of  5  months  but   10  miles  of  the  road  were  built :  how 
many  more  men  must  be  employed  to  finish  the  road  in  the 
time  agreed  upon  ? 

18.  If  336  men,  in  5  days  of  10  hours  each,  can  dig  a  trench 
of  5  degrees  of  hardness,  70  yards  long  3  wide  and  2  deep  : 
what  length  of  trench  of  6  degrees  of  hardness,  5  yards  wide 
and  3  yards  deep,  may  be  dug  by  240  men  in  9  days  of  12 
hours  each  ? 


228  PARTNERSHIP. 


PARTNERSHIP. 

240.  PARTNERSHIP  is  the  joining  together  of  two  or  more 
persons  in  trade,  with  an  agreement  to  share  the  profits  or 
losses. 

PARTNERS  are  those  who  are  united  together  in  carrying 
on  business. 

CAPITAL,  is  the  amount  of  money  or  property  employed : 
DIVIDEND  is  the  gain  or  profit : 
Loss  is  the  opposite  of  profit : 

241.  The  Capital  or  Stock  is  the  cause  of  the  entire  profit : 
Each  man's  capital  is  the  cause  of  his  profit : 

The  entire  profit  or  loss  is  the  effect  of  the  whole  capital : 
Each  man's  profit  or  loss  is  the  effect  of  his  capital :  hence, 

Wliole  Stock  :  Each  man's  Stock 
:  :  Whole  profit  or  loss  :  Each  man's  profit  or  loss. 

EXAMPLES. 

1.  A  and  B  buy  certain  goods  amounting  to  160  dollars,  of 
which  A  pays  90  dollars  and  B,  70  ;  they  gain  32  dollars  by 
the  purchase  :  what  is  each  one's  share  ? 

OPERATION. 

160  :  90  :  :  32  :  A's  share  ;  or, 


160  :  :  70  :  32  :  B's  share  ;  or, 


240.  What  is  a  partnership  ?    What  are  partners  ?    What  is  capital 
or  stock  ?     What  is  dividend  ?    What  is  loss  ? 

241.  What  is  the  cause  of  the  profit?    What  is  the  cause  of  each 
man's  profit?    What  is  the  effect  of  the  whole  capital  ?    What  is  the 
effect  of  each  man's  capital  ?    What  proportion  exists  between  causes 
and  their  effects  ?    What  is  the  rule  ? 


COMPOUND   PARTNERSHIP.  229 

Hence,  the  following 

RULE. — As  the  whole  stock  is  to  each  man's  share,  so  is  the 
whole  gain  or  loss  to  each  man's  share  of  the  guin  or  loss. 

EXAMPLES. 

1.  A  and  B  have  a  joint  stock  of  $2100,  of  which  A  owns 
$1800  and  B  $300  ;  they  gain  in  a  year  $1000  :  what  is 
each  one's  share  of  the  profits  ? 

2.  A,  B  and  C  fit  out  a  ship  for  Liverpool.     A  contributes 
$3200,  B  $5000,  and  C  $4500  ;  the  profits  of  the  voyage 
amount  to  $1905  :  what  is  the  portion  of  each  ? 

3.  Mr.  Wilson  agrees  to  put  in  5  dollars  as  often  as  Mr. 
Jones  puts  in  7  ;  'after  raising  their  capital  in  this  way,  they 
trade  for  1  year  and  find  their  profits  to  be  $3600  :  what  is 
the  share  of  each  ? 

4.  A.  B  and  C  make  up  a  capital  of  $20,000  ;  B  and  C 
each  contribute  twice  as  much  as  A  ;  but  A  is  to  receive  one- 
third  of  the  profits  for  extra  services  ;  at  the  end  of  the  year 
they  have  gained  $4000  :  what  is  each  to  receive  ? 

5.  A,  B  and  C  agree  to  build  a  railroad  and  contribute 
$18000  of  capital,  of  which  B  pays  2  dollars,  and  C,  3  dollars 
as  often  as  A  pays  1  dollar  ;  they  lose  $2400  by  the  opera- 
tion :  what  is  the  loss  of  each  ? 

COMPOUND  PARTNERSHIP. 
242.   When  the  causes  of  profit  or  loss  are  compound. 

"When  the  partners  employ  their  capital  for  different  periods 
of  time,  each  cause  of  profit  or  loss  is  compound,  being  made 
up  of  the  two  elements  of  capital  and  t^me.  The  product  of 
these  elements,  in  each  particular  case,  will  be  the  cause  of 
each  man's  gain  or  loss  ;  and  their  sum  will  be  the  cause  of 
the  entire  gain  or  loss  :  hence,  to  find  each  share, 

Multiply  each  man1  stock  by  the  time  he  continued  it  in 
trade  ;  then  say,  as  the  sum  of  the  products  is  to  each  product, 
so  is  the  whole  gain  or  loss  to  each  man's  share  of  the  gain  or 


243.  "When  is  the  cause  of  profit  or  loss  compound  ?    What  arc  the 
elements  of  the  compound  caus  •?     What  is  the  rule  in  this  case? 


230  COMPOUND   PARTNERSHIP. 


EXAMPLES. 

1.  A  and  B  entered  into  partnership.     A  put  in  $840  for  4 
months,  and  B,  $650  for  6  months  ;  they  gained  $363  :  what 
is  each  one's  share  ? 

OPERATION. 

A,  $840x4-3360 

B.  650  x  6—3900 

J3360  :  :  QPQ       f  $168  A's. 
J3900   ::  363:    j  $195  B's. 

2.  A  puts  in  trade  $550  for  7  months  and  B  puts  in  $1625 
for  8  months  ;  they  make  a  profit  of  $337  :  what  is  the 
share  of  each  ? 

3.  A  and  B  hires  a  pasture,  for  which  they  agreed  to  pay 
$92.50.     A  pastures  12  horses  for  9  weeks  and  B  11  horses 
for  7  weeks  :  what  portion  must  each  pay  ? 

4.  Four  traders  form  a  company.     A  puts  in  $400  for  ft 
months  ;  B  $600  for  7  months  ;  C  $960  for  8  months  ;    D 
$1200  for  9  months.     In  the  course  of  trade  they  lost  $750  ; 
how  much  falls  to  the  share  of  each  ? 

5.  A,  B  and  C  contribute  to  a  capital  of  $15000  in  the 
following  manner :  every  time  A  puts  in  3  dollars  B  puts  in 
$5  and  C,  $7.     A's  capital  remains  in  trade  1  year  ;  B's  If- 
years  ;  and  C's  2f  years  ;  at  the  end  of  the  time  there  is  a 
profit  of  $15000  :  what  is  the  share  of  each  ? 

6.  A  commenced  business  January  1st,  with  a  capital  of 
$3400.     April  1st,  he  took  B  into  partnership,  with  a  capital 
of  $2600  ;  at  the  expiration   of  the  year  they  had  gained. 
$750  :  what  is  each  one's  share  of  the  gain  ? 

7.  James  Fuller,  John  Brown  and  William  Dexter  formed 
a  partnership,  under  the  firm  of  Fuller,  Brown  &  Co.,  with  a 
capital  of  $20000  ;  of  which  Fuller  furnished  $6000,  Brown 
$5000,  and  Dexter  $9000.     At  the  expiration  of  4  months, 
Fuller  furnished  $20^)0  more  ;  at  the  expiration  of  6  months, 
Brown  furnished  $2500  more  ;  and  at  the  end  of  a  year  Dex- 
ter withdrew  $2000.     At  the  expiration  of  one  year  and  a 
half,  they  found  their  profits  amounted  to  $5400  :  what  was 
each  partner's  share  ? 


PERCENTAGE. 


231 


PERCENTAGE. 

243.  PERCENTAGE  is  an  allowance  made  by  the  hundred. 
The  base  of  percentage,  is  the  number  on  which  the  per- 
centage is  reckoned. 

PER  CENT  means  by  the  hundred  :  thus,  1  per  cent  means 

1  for  every  hundred  ;  2  per  cent,  2  for  every  hundred  ;  3  per 
cent,  3  for  every  hundred,  &c.     The  allowances,  1  per  cent, 

2  per  cent,  3  per  cent,  &c.,  are  called  rates,  and  may  be 
expressed  decimally,  as  in  the  following 

TABLE. 


1  per  cent  is 

-01 

7  per  cent  is 

.07 

3  per  cent  is 

.03 

3  per  cent  is 

.08 

4  per  cent  is 

.04" 

15  per  cent  is 

.15 

5  per  cent  is 

.05 

68  per  cent  is 

.68 

6  percent  is 

.06 

99  per  cent  is 

.99 

100  per  cent  is  1. 
150  per  cent  is  1.50 
130  per  cent  is  1.30 
200  per  cent  is  2. 
.  \  per  cent  is  .005 
3|  per  cent  is  .035 
5|  per  cent  is  0575 


ALSO, 

for,  1-0$  is  equal  to  1 . 
for,  |£g  is  equal  to  1.50 
for,  |$#  is  equal  to  1.30 
for,  f  $£  is  equal  to  2.00 
for,  T-^-^2  is  equal  to  .005 
for,  3J  =  .03+.005  =  .035 
for,  5j=.05+.075  =  .OT5 


EXAMPLES. 

Write,  decimally,  8J  per  cent  ;  9  per  cent  ;  6|  per  cent  ; 
65J  per  cent  ;  205  per  cent  ;  327  per  cent. 

244.   To  find  the  percentage  of  any  number. 

1.  What  is  the  percentage  of  $320,  the  rate  being  5  per 
cent? 


343.  What  is  per  centage?  What  is  the  base?  What  does  per  cent 
mean  ?  What  do  you  understand  by  3  per  cent  ?  What  is  the  rate,  or 
rate  per  cent  ? 

244.  How  do  yon  find  the  percentage  of  any  number  ? 


232  PERCENTAGE. 

ANALYSIS. — The  rate  being  5  per  cent,  is  ex-  OPERATION. 
pressed  decimally  by  .05.     We  are  then  to  take  320 

.05  of  the  base  (which  is  $320) ;  this  we  do  by 
multiplying  $320  by  .05. 

Hence,  to  find  the  percentage  of  a  number,  $16. 00  Ans. 

Multiply  the  number  by  the  rate  oppressed  decimally,  and 
the  product  will  be  the  percentage. 

EXAMPLES. 

1.  What  is  the  percentage  of  $657,  the  rate  being  4J  per 
cent? 

OPERATION 

NOTE.— When   the    rate   cannot   be  .657 

reduced  to  an  exact  decimal,  it  is  most  Q^I 

convenient  to  multiply  by  the  fraction, — 

and  then  by  that  part  of  the  rate  which  219  =  |  per  cent, 

is  expressed  in  exact  decimals.  2628  =  4  per  cent. 

$28.47  =  41  per  cent. 
Find  the  percentage  of  the  following  numbers  : 


1.  2J  per  cent  of  650  dollars. 

2.  3  per  cent  of  650  yards. 

3.  4  £  per  cent  of  Slbcwl. 

4.  6J  per  cent  of  $37.50. 

5.  5|  per  cent  of  2704  miles. 

6.  \  per  cent  of  1000  oxen. 
7  2|  per  cent  of  $376. 

8.  2^  per  cent  of  860  sheep. 

9.  5§  per  cent  of  $327.33. 


10.  66|  per  cent  of  420  cows. 

11.  105  per  cent  of  850  tons. 

12.  116  per  cent  of  875/6. 

13.  241  per  cent  of  $875.12. 

14.  37J  per  cent  of  $200. 

15.  33^  per  cent  of  $687.24. 

16.  87J  per  cent  of  $400. 

17.  62J  per  cent  of  $600. 

18.  308  per  cent  of  $225.40. 


19.  A  has  $852  deposited  in  the  bank,  and  wishes  to  draw 
out  5  per  cent  of  it :  how  much  must  he  draw  for  ? 

20.  A  merchant  has   1200  barrels  of  flour  :  he  shipped 
64  per  cent  of  it  and  sold  the  remainder  :  how  much  did  he 
sell? 

21.  A  merchant  bought  1200  hogsheads  of  molasses.     On 
getting  it  into  his  store,  he  found  it  short  3|  per  cent :  how 
many  hogsheads  were  wanting  ? 

122.  What  is  the  difference  between  5|  per  cent  of  $800 
and  6J  per  cent  of  $1050? 


PERCENTAGE.  233 

23.  Two  men  had  each  $240.     One  of  them  spends  14 
per  cent,  and  the  other  18|  per  cent :  how  many  dollars  more 
did  one  spend  than  the  other  ? 

24.  A  man  has  a  capital  of  $12500  :  he  puts  15  per  cent 
of  it  in  State  Stocks  :  33 J  per  cent  in  Railroad  Stocks,  and 
25  per  cent  in  bonds  and  mortgages  :  what  per  cent  has  he 
left,  and  what  is  its  value  ? 

25.  A  farmer  raises  850  bushels  of  wheat :  he  agrees  to 
sell  18  per  cent  of  it  at  $1.25  a  bushel ;  50  per  cent  of  it  at 
$1.50  a  bushel,  and  the  remainder  at  $1.75  a  bushel :  how 
much  does  he  receive  in  all  ? 

245.  To  find  the  per  cent  which  one  number  is  of  another. 
1.  What  per  cent  of  $16  is  $4  ? 

ANALYSIS. — The  question  is,  what  part  of  OPERATION. 

$16  is  $4,  when  expressed  in  hundreths:  JL-— 1  — .25. 

The  standard  is  $16  (Art.  228) :  hence,  the       or  25  per  cent, 
part  is  -j*g±:^— .25;  therefore,  the  per  cent  is 
25  :  hence,  to  find  what  per  cent  one  number  is  of  another, 

Divide  by  the  standard  or  base,  and  the  quotient,  reduced 
to  decimals,  will  express  the  rate  per  cent. 

NOTE. — The  standard  or  base,  is  generally  preceded  by  the  word 
of. 

EXAMPLES. 

1.  What  per  cent  of  20  dollars  is  5  dollars? 

2.  Forty  dollars  is  what  per  cent  of  eighty  dollars  ? 

3.  What  per  cent  of  200  dollars  is  80  dollars  ? 

4.  What  per  cent  of  1250  dollars  is  250  dollars  ? 

5.  What  per  cent  of  650  dollars  is  250  dollars  ? 

6.  Ninety  bushels  of  wheat  is  what  per  cent  of  ISOO&usJi.? 

7.  Nine  yards  of  cloth  is  what  per  cent  of  870  yards  ? 

8.  Forty-eight  head  of  cattle  are  what  per  cent  of  a  drove 
of  1600  ? 

9.  A  man  has  $550,  and  purchases  goods  to  the  amount 
of  $82.75  :  what  per  cent  of  his  money  does  he  expend? 

245.  How  do  you  find  the  per  cent  which  one  number  is  of  another  ? 


234  PERCENTAGE. 

10.  A  merchant  goes  to  New  York  with  $1500  ;  he  first 
lays  out  20  "per  cent,  after  which  he  expends  $660  :  what 
per  cent  was  his  last  purchase  of  the  money  that  remained 
after  his  first  ? 

11.  Out  of  a  cask  containing  300  gallons,  60  gallons  are 
drawn  :  what  per  cent  is  this  ? 

12.  If  I  pay  $698.23  for  3  hogsheads  of  molasses  and  sell 
them  for  $837.996,  how  much  do  I  gain  per  cent  on  the 
money  laid  out  ? 

13.  A  man  purchased  a  farm  of  75  acres  at  $42.40  an 
acre.     He  afterwards  sold  the  same  farm  for  $3577.50  :  what 
was  his  gain  per  cent  on  the  purchase  money  ? 

STOCK,  COMMISSION  AND  BROKERAGE. 

246.  A  CORPORATION  is  a  collection  of  persons  authorized 
by  law  to  do  business  together.     The  law  which  defines  their 
rights  and  powers  is  called  a  Charter. 

CAPITAL  or  STOCK  is  the  money  paid  in  to  carry  on  the 
business  of  the  Corporation,  and  the  individuals  so  contributing 
are  called  Stockholders.  This  capital  is  divided  into  equal 
parts  called  Shares,  and  the  written  evidences  of  ownership 
are  called  Certificates. 

247.  When  the  United  States  Government,  or  any  of  the 
States,  borrows  money,  an  acknowledgment  is  given  to  the 
lender,  in  the  form  of  a  bond,  bearing  a  fixed  interest.     Such 
bonds  are  called  United  States  Stock,  or  State  Stock. 

The  par  value  of  stock  is  the  number  of  dollars  named  in 
each  share.  The  market  value  is  what  the  stock  brings  per 
share  when  sold  for  cash. 

If  the  market  value  is  above  the  par  value,  the  stock  is 
said  to  be  at  a  premium,  or  above  par ;  but  if  the  market 
value  is  below  the  par  value,  it  is  said  to  be  at  a  discount,  or 
below  par. 

346.  What  is  a  corporation  ?    What  is  a  charter?    What  is  capital 
or  stock  ?    What  are  shares  ? 

347.  What  are  United  States  Stocks?     What    are   State   Stocks? 
What  is  the  par  value  of  a  stock  ?    What  is  the  market  value  ?    If  the 
market  is  above  the  par  value,  what  is  said  of  the  stock  ?    If  it  is  below, 
what  is  said  of  the  stock  ?    What  is  the  market  value  when  above  par  ? 
What  when  below  ? 


COMMISSION   AND   BROKERAGE.  235 

Let  l=par  value  of  1  dollar  : 

l+premium= market  value  of  1  dollar, 'when  above 

par  : 
1 — discount  =:  market  value  of  1  dollar  when  below  par. 

248.  Commission  is  an  allowance  made  to  an  agent  for 
buying  or  selling,  or  taking  charge  of  property,  and  is  gen- 
erally reckoned  at  a  certain  rate  per  cent. 

The  commission,  for  the  purchase  or  sale  of  goods  in  the 
city  of  New  York,  varies  from  2J  to  12  J  per  cent,  and  under 
some  circumstances  even  higher  rates  are  paid. 

Brokerage  is  an  allowance  made  to  an  agent  who  buys  or 
sells  stocks,  uncurrent  money,  or  bills  of  exchange,  and  is 
generally  reckoned  at  so  much  per  cent  on  the  par  value  of 
the  stock.  The  brokerage,  in  the  city  of  New  York,  is  gene- 
rally one-fourth  per  cent  on  the  par  value  of  the  stock. 

EXAMPLES. 

1.  What  is  the  commission  on  $4396  at  per  6  cent? 

OPERATION. 

NOTE. — We  here  find  the   commission,  as  $4396 

in  simple  percentage,  by  multiplying  by  the  de-  Q  g 

cimal  which  expresses  the  rate  per  cent.  : 

Am.  $263.76. 

2.  A  factor  sells  60  bales  of  cotton  at  $425  per  bale,  and 
is  to  receive  2  J  per  cent  commission  :  how  much  must  he  pay 
over  to  his  principal  ? 

3.  A  drover  agrees  to  purchase  a  drove  of  cattle  and  to  sell 
them  in  New  York  city  for  5  per  cent  on  what  he  may  re- 
ceive ;  he  expends  in  the  purchase  $4250,  and  sells  them  at 
an  advance  of  10  per  cent :  how  much  is  his  commission  ? 

4.  A  commission  merchant  sells  goods  to  the  amount  of 
$8750,  on  which  he  is  to  be  allowed  2  per  cent,  but  in  con- 
sideration of  paying  the  money  over  before  it  is  due,  he  is  to 
receive  !-£•  per  cent  additional :  how  much  must  he  pay  over 
to  his  principal  ? 

5.  A  broken  bank  has  a  circulation  of  $98000  and  pur- 
chases the  bills  a,t  85  per  cent :  how  much  is  made  by  the 
operation  ? 

248.  What  is  commission  ?    What  is  brokerage  ? 


236  PERCENTAGE. 

6.  Merchant  A  sent  to  B,  a  broker,  $3825  to  be  invested  in 
stock  ;  B  is  to  receive  2  per  cent  on  the  amount  paid  for  the 
stock  :  what  was  the  value  of  the  stock  purchased  ? 

OPERATION. 

ANALYSTS.— Since  the  broker  re-  1 .02)3825  .00($3750vl?is. 

ceives  2  per  cent,  it  will    require  306 

$1.02  to  purchase  1  dollar's  worth  

of   stock;  hence,  there  will   be  as  765 

many  dollar's  worth    purchased  as  714 
$1.02  is  contained  times  in  $3825  ; 
that  is,  $3750  worth. 

510 

7.  Mr.  Jones  sends  his  broker  $18560  to  be  invested  in 
U.  S.  Stocks,  which  are  15  per  cent  above  par  ;  the  broker  is 
to  receive  one  per  cent  ;  how  many  shares  of  $100  each  can 
be  purchased  ? 

ANALYSIS. — Since  the  premium  is  15 
per  cent,  and  the  brokerage  1  per  cent,  OPERATION. 

each  dollar  of  par  value  will  cost  $1       1.16)18560 
plus  the  premium,  plus  the  brokerage^ 

$1.16  :   hence,  the  amount    purchased  '°  quotient, 

wiU  be  as  many  dollars  as   $1.16  is     or,       160       shares, 
contained  times  in  $18560. 

8.  I  have  $4999.89  to  be  laid  out  in  stocks,  which  are  15 
per  cent  below  par  :  allowing  2  per  cent  commission,  how 
much  can  be  purchased  at  the  par  value  ? 

ANALYSIS. — Since   the  stock  is  at  a  dis- 
count of  15  per  cent,  the  market  value  will         OPERATION. 
be  85  per  cent ;  add  2  per  cent,  the  broker-      .  87)4999.89 
age,  gives  87  per  cent=.87.     The  amount  v-^  . ^ — . 

purchased  will  be  as  many  dollars  as  .87  is 
contained  times  in  $4999,89. 

Hence,  to  find  the  amount  at  par  value, 

Divide  the  amount  to  be  expended  by  the  market  value  of 
$1  plus  the  brokerage  ;  and  the  quotient  ivill  be  the  amount 
in  par  value. 

9.  Messrs.  Sherman  &  Co. received  of  Mr  Gilbert  $28638.50 
to  be  invested  in  bank  stocks,  which  are  12i  per  cent  above 
par,  for  which  they  are  to  receive  one-fourth  of  one  per  cent 
commission  :  how  many  shares  of  $127  each  can  they  buy  ? 


LOSS   OR  GAIN.  237 

10.  The  par  value  of  Illinois  Railroad  stock  is  100.  It 
sells  in  market  at  72 J  :  if  I  pay  J  per  cent  brokerage,  how 
many  shares  can  I  buy  for  $5820  ? 

PROFIT  AND  LOSS. 

249.  Profit  or  loss  is  a  process  by  which  merchants  dis- 
cover the  amount  gained  or  lost  in  the  purchase  and  sale  of 
goods.      It  also  instructs   them  how  much   to  increase  or 
diminish  the  price  of  their  goods,  so  as  to  make  or  lose  so 
much  per  cent. 

EXAMPLES. 

1.  Bought  a  piece  of  cloth  containing  75?/d.  at  $5.25  per 
yard,  and  sold  it  at  $5.75  per  yard  :  how  much  was  gained 
in  the  trade  ? 

OPERATION. 

ANALYSIS.— We    first    find    tho     $5.75  price  of  1  yard, 
profit  on  a  single  yard,  and  then     AC  op;  oc.^  nf  i  vnrfi 
multiply  by  the  number  of  yards,     !£^_co 
which  is  *5.  50cfe.  profit  on  1  yard  : 

then,  $0.50x75=$37.50. 

2.  Bought  a  piece  of  calico  containing  56  yards,  at  27  cents 
a  yard  :  what  must  it  be  sold  for  per  yard  to  gain  $2.24  ? 

OPERATION. 

56  yards  at  27  cents=$15.12 

ANALYSIS.— First   find   the      Profit  -  2.24 

cost,  then  add  the  profit  and       T,  ,,   ,, 

divide  the  sum  by  the  number       Ifc  must  sel1  f°r     '          WM. 
of  yards  56)17,36 

31  cts.  a  yard. 

250.  Knowing  the  per  cent,   of  gain  or  loss   and  the 
amount  received,  to  find  the  cost. 

1.  I  sold  a  parcel  of  goods  for  $195.50,  on  which  I  made 
15  per  cent :  what  did  they  cost  me  ? 

ANALYSIS. — 1  dollar  of  the  cost  plus  15  per  OPERATION. 

cent,  will  be  what  that  which  cost  $1  sold  for,  1.15)  195.50 

viz ,    $1.15  :    hence,  there  will    be   as   many  ^   K  — — 

dollars  of  cost,  as  $1.15  is  contained  times  in  *L1()  Ans- 
what  the  goods  brought. 

349.  What  is  loss  or  gain  ? 


238  PERCENTAGE. 

2.  If  I  sell  a  parcel  of  goods  for  $170,  by  which  I  lose 
15  per  cent,  what  did  they  cost  ? 

ANALYSIS. — 1  dollar  of   the  cost  less  15  per        OPERATION. 
cent,  will  be  what  that  which  cost  1  dollar  sold          .85)  170 
for,  viz.,  $0.85  :  hence,  there  will  be  as  many 
dollars  of  cost,   as   .85   is  contained  times  in 
what  the  goods  brought. 

Hence,  to  find  the  cost, 

Divide  the  amount  received  by  1  plus  the  per  cent  ivhen 
there  is  a  gain,  and  by  1  minus  the  per  cent  when  there 
is  a  loss,  and  the  quotient  will  be  the  cost. 

EXAMPLES. 

1.  Bought  a  piece  of  cassimere  containing  28  yards  at 
1  £  dollars  a  yard  ;  but  finding  it  damaged,  am  willing  to  sell 
it  at  a  loss  of  15  per  cent :  how  much  must  be  asked  per 
yard? 

2.  Bought  a  hogshead  of  brandy  at  $1.25  per  gallon,  and 
sold  it  for  $78  :  was  there  a  loss  or  gain  ? 

3.  A  merchant  purchased  3275  bushels  of  wheat  for  which 
he  paid  $3517.10,  but  finding  it  damaged,  is  willing  to  lose 
10  per  cent :  what  must  it  sell  for  per  bushel  ? 

4.  Bought  a  quantity  of  wine  at  $1.25  per  gallon,  but  it 
proves  to  be  bad  and  am  obliged  to  sell  it  at  20  per  cent  less 
than  I  gave  :  how  much  must  I  sell  it  for  per  gallon  ? 

5.  A  farmer  sells  125  bushels  of  corn  for  75  cents  per 
bushel ;  the  purchaser  sells  it  at  an  advance  of  20  per  cent : 
how  much  did  he  receive  for  the  corn  ? 

6.  A  merchant  buys  1  tun  of  wine  for  which  he  pays  $725, 
and  wishes  to  sell  it  by  the  hogshead  at  an  advance  of  15  per 
cent :  what  must  be  charged  per  hogshead  ? 

7.  A  merchant  buys  158  yards  of  calico  for  which  he  pays 
20  cents  per  yard  ;  one-half  is  so  damaged  that  he  is  obliged 
to  sell  it  at  a  loss  of  6  per  cent :  the  remainder  he  sells  at  an 
advance  of  19  per  cent :  how  much  did  he  gain? 

8.  If  I  buy  coffee  at  16  cents  and  sell  it  at  20  cents  a 
pound,  how  much  do  I  make  per  cent  on  the  money  paid  ? 

250.  Knowing  the  per  cent  of  gain  or  loss  and  the  amount  received 
how  do  you  find  the  cost  ? 


INSURANCE.  i!39 

9.  A  man  bought  a  house  and  lot  for  $1850.50,  and  sold  it 
for  $1517.41  :  how  much  per  cent  did  he  lose  ? 

10.  A  merchant  bought  650  pounds  of  cheese  at  10  cents 
per  pound,  and  sold  it  at  12  cents  per  pound  :  how  much  did 
he  gain  on  the  whole,  and  how  much  per  cent  on  the  money 
laid  out  ? 

11.  Bought  cloth  at  $1.25  per  yard,  which  proving  bad,  I 
wish  to  sell  it  at  a  loss  of  18  per  cent :  how  much  must  I 
ask  per  yard  ? 

12.  Bought  50  gallons  of  molasses  at  75  cents  a  gallon, 
10  gallons  of  which  leaked  out.     At  what  price  per  gallon 
must  the  remainder  be  sold  that  I  may  clear  10  per  cent  on 
the  cost  ? 

13.  Bought  67  yards  of  cloth  for  $112,  but  19  yards  being 
spoiled,  I  am  willing  to  lose  5  per  cent :  how  much  must  I 
sell  it  for  per  yard  ? 

14.  Bought  67  yards  of  cloth  for  $112,  but  a  number  of 
yards  being  spoiled,  I  sell  the  remainder  at  $2.216|  per  yard, 
and  lose  5  per  cent :  how  many  yards  were  spoiled  ? 

15.  Bought  2000  bushels  of  wheat  at  $1.75  a  bushel,  from 
which  was  manufactured  475  barrels  of  flour  :  what  must 
the  flour  sell  for  per  barrel  to  gain  25  per  cent  on  the  cost  of 
the  wheat  ? 

INSURANCE. 

251.  INSURANCE  is  an  agreement,  generally  in  writing,  by 
which  an  individual  or  company  bind  themselves  to  exempt 
the  owners  of  certain  property,  such  as  ships,  goods,  houses, 
&c.,  from  loss  or  hazard. 

The  POLICY  is  the  written  agreement  made  by  the  parties. 

PREMIUM  is  the  amount  paid  by  him  who  owns  the  property 
to  those  who  insure  it,  as  a  compensation  for  their  risk.  The 
premium  is  generally  so  much  per  cent  on  the  property  in- 
sured. 

EXAMPLES. 

1.  What  would  be  the  premium  for  the  insurance  of  a 
house  valued  at  $8754  against  loss  by  fire  for  one  year,  at 
\  per  cent  ? 

251.  What  is  insurance?  What  is  the  policy?  What  is  the  pre- 
mium ?  How  is  it  reckoned  ? 


PERCENTAGE. 

2.  What  would  bo  the  premium  for  insuring  a  ship  and 
cargo,  valued  at  $37500,  from  New  York  to  Liverpool,  at  3£ 
per  cent  ? 

3.  What  would  be  the  insurance   on   a   ship   valued  at 
$47520  at  J  per  cent ;  also  at  J  per  cent? 

4.  What  would  be  the  insurance  on  a  house  valued  at 
$14000  at  1J  per  cent? 

5.  What  is  the  insurance  on  a  store  and  goods  valued  at 
$27000,  at  2  J  per  cent  ? 

6.  What  is  the  premium  of  insurance  on  $9870  at  14  per 
cent? 

7.  A  merchant  wishes  to  insure  on  a  vessel  and  cargo  at 
sea,  valued  at  $28800  :  what  will  be  th^  premium  at  1|  per 
cent  ? 

8.  A  merchant  owns  three-fourths  of  a  ship  valued  at 
$24000,  and  insures  his  interest  at  2|  per  cent :  what  does 
he  pay  for  his  policy  ? 

9.  A  merchant  learns  that  his  vessel  and  cargo,  valued 
at  $36000,  have  been  injured  to  the  amount  of  $12000  ;  he 
effects  an  insurance  on  the  remainder  at  5|  per  cent ;  what 
premium  does  he  pay  ? 

10.  My  furniture,  worth  $3440,  is  insured  at  2f  per  cent ; 
my  house,  worth  $1000,  at  1 J  per  cent ;  and  my  barn,  horses 
and  carriages,  worth  $1500,  at  3J  per  cent :  what  is  the 
whole  amount  of  my  insurance  ? 

11.  A  man  bought  a  house,  and  paid  the  insurance  at  2| 
per  cent,  the  whole  of  which  amounted  to  $1845  :  what  was 
the  value  of  the  house  and  the  amount  of  the  insurance  ? 

12.  What  would  it  cost  to  insure  a  store,  worth  $3240,  at 
f  per  cent,  and  the  stock,  worth  $7515.75,  at  f  per  cent? 

13.  A  merchant  imported  250  pieces  of  broadcloth,  each 
piece  containing  36|  yards,  at  $3.25  cents  a  yard.     He  paid 
4|  per  cent  insurance  on  the  selling  price,  $4.50  a  yard.     If 
the  goods  were  destroyed  by  fire,  and  he  got  the  amount  of 
insurance,  how  much  did  he  make  ? 

14.  A  vessel  and  cargo,  worth  $65000,  are  damaged  to  the 
amount  of  20  per  cent,  and  there  is  an  insurance  of  50  per 
cent  on  the  loss:  how  much  insurance  will  the  owner  re- 
ceive ? 


INTEREST.  241 


INTEREST. 

252.  INTEREST  is  an  allowance  made  for  the  use  of  money 
that  is  borrowed. 

PRINCIPAL  is  the  money  on  which  interest  is  paid. 
AMOUNT  is  the  sum  of  the  Principal  and  Interest. 
For  example  :    If  I  borrow  1  dollar  of  Mr.  Wilson  for  1 
year,  and  pay  him  7  cents  for  the  use  of  it ;  then, 

1  dollar  is  the  principal, 

7  cents  is  the  interest,  and 

$1.07  the  amount. 

The  RATE  of  interest  is  the  number  of  cents  paid  for  the 
use  of  1  dollar  for  1  year.  Thus,  in  the  above  example,  th*e 
rate  is  7  per  cent  per  annum. 

NoTE.-VThe  term  per  cent  means,  ty  the  hundred;  and  per 
annum  means  by  the  year.  As  interest  is  always  reckoned  by  the 
year,  the  term  per  annum  is  understood  and  omitted. 

CASE  I. 

253.  To  find  the  interest  of  any  principal  for  one  or  more 
years. 

1.  What  is  the  interest  of  $1960  for  4  years,  at  7  per 
cent? 

ANALYSIS. — The   rate  of  interest 

being  7  per  cent,  is  expressed  deci-  OPERATION. 

mallyby.07:  hence,  each  dollar,  in  $1960 

1  year  will  produce  .07  of  itself,  and  A  7  rqfp 

$1960  will    produce   .07  of   $1960,         — 

or  $137.20.    Therefore,  $137.20  is  the  137.20  int.  for  It/r. 

interest  for  1  year,  and  this  interest  4  No.  of  years, 

multiplied  by  4,  gives  the  interest  for  AC  4Q  Qft 

4  years :  hence,  the  following  $D48.»U 

RULE. — Multiply  the  principal  by  the  rate,  expressed 
decimally,  and  the  product  by  the  number  of  years. 

252.  What   is   interest?     What  is   principal?     What   is    amount? 
What  is  rate  of  interest  ?    \Vhat  does  per  annum  mean  ? 

253.  How  do  you  find  the  interest  of  any  principal  for  any  number  of 
years  ?    Give  the  analysis. 


242  SIMPLE    INTEREST. 

EXAMPLES. 

1.  What  is  the  interest  of  $365.874  for  one  year,  at  5J 
per  cent  ? 

OPERATION. 

365.874 

ANALYSIS. — We  first  find  the  in-  951 

terest  at  ^  per  cent,  and  then  the          — — - — — 
interest  at  5  per  cent ;  the  sum  is  1.82937  £  per  cent, 

the  interest  at  5£  per  cent.  18.29370  5  per  cent. 

Ans.  $20.12307  5J  per  cent. 

2.  What  is  the  interest  of  $650  for  one  year,  at  6  per  cent  ? 

3.  What  is  the  interest  of  $950  for  4  years,  at  7  per  cent  ? 

4.  What  is  the  amount  of  $3675  in  3  years,  at  7  per  cent  ? 

5.  What  is  the  amount  of  $459  in  5  years,  at  8  per  cent  ? 

6.  What  is  the  amount  of  $375  in  2  years,  at  7  per  cent? 

7.  What  is  the  interest  of  $21 1.26  for  1  year,  at  4J  per  ct.  ? 

8.  What  is  the  interest  of  $1576.91  for  3  years,  at  7  per  ct.  ? 

9.  What  is  the  amount  of  $957.08  in  6  years,  at  3J  per  ct.  ? 

10.  What  is  the  interest  of  $375.45  for  7  years,  at  7  per  ct.  ? 

11.  What  is  the  amount  of  $4049.87  in  2  years,  at  5  per  ct.  ? 

12.  What  is  the  amount  of  $16199.48  in  16  yrs.,  at  5J  per  ct.  ? 

NOTE. — When  there  are  years  and  months,  and  the  months  are 
aliquot  parts  of  a  year,  multiply  the  interest  for  1  year  by  the  years 
and  months  reduced  to  the  fraction  of  a  year. 

EXAMPLES. 

1.  What   is   the   interest  of    $326.50,  for  4   years    and 

2  months,  at  7  per  cent  ? 

2.  What   is   the   interest   of  $437.21,    for    9   years   and 

3  months,  at  3  per  cent  ? 

3.  What  is  the  amount  of  $1119.48,  after  2  years  and 
6  months,  at  7  per  cent  ? 

4.  What  is  the  amount  of  $179.25,   after  3  years  and 

4  months,  at  7  per  cent? 

5.  What  is  the  amount  of  $1046.24,  after  4  years  and 
3  months  at  5^  per  cent  ? 


SIMPLE   INTEREST.  24:3 


CASE    II. 

254.  To  find  the  interest  on  a  given  principal  for  any  rate 
and  time. 

1.  What  is  the  interest  of  $876.48  at  6  per  cent,  for 
4  years  9  months  and  14  days  ? 

ANALYSIS. — The  interest  for  1  year  is  the  product  of  the  princi- 
pal multiplied  by  the  rate  If  the  interest  for  1  year  be  divided 
by  12,  the  quotient  will  be  the  interest  for  1  month  :  if  the  interest 
for  1  month  be  divided  by  30,  the  quotient  will  be  the  interest 
for  1  day. 

The  interest  for  4  years  is  4  times  the  interest  for  1  year ;  the 
interest  for  9  months,  9  times  the  interest  for  1  month  ;  and  the 
interest  for  14  days,  14  times  the  interest  for  1  day 

OPERATION. 

$876.48 
.06 


12)52.5888=int.  for  lyr.     52.5888    x    4 =$210.3552  4yr. 
30)4.3824 =int.  for  Imo.      4.3824    x    9  =  $  39.4416  9mo. 
.14608=int.  for  Ida.         .14608  x  14=$     2.0451  Udg. 
Total  interest,  $251.84194- 

Hence,  we  have  the  following 

RULE. — I.  Find  the  interest  for  1  year : 

II.  Divide  this  interest  by  12,  and  the  quotient  will  be  the 
interest  for  1  month  : 

III.  Divide  the  interest  for  1  month  by  30,  and  the  quo- 
tient will  be  the  interest  for  1  day. 

IY.  Multiply  the  interest  for  1  year  by  the  number  of 
years,  the  interest  for  1  month  by  the  number  of  months,  and 
the  interest  for  1  day  by  the  number  of  days,  and  the  sum 
of  the  product  will  be  the  required  interest. 

NOTE. — In  computing  interest  the  month  is  reckoned  at  30  days. 

2.  What  is  the  interest  of  $132.26  for  1  year  4  months 
and  10  days,  at  6  per  cent  per  annum  ? 

3  What  is  the  interest  of  $25.50  for  1  year  9  months  and 
12  days,  at  6  per  cent  ? 

254.  How  do  you  find  the  interest  for  any  time  at  any  rate  ? 


244:  SIMPLE   INTEREST. 

^  2D  METHOD. 

255.  There  is  another  rule  resulting  from  the  last  analysis, 
which  is  regarded  as  the  best  general  method  of  computing 
interest. 

RULE. — I.  Find  the  interest  for  1  year  and  divide  it  bylZ: 
the  quotient  will  be  the  interest  for  1  month. 

II.  Multiply  the  interest  for  1  month  by  the  time  expressed 
in  months  and  parts  of  a  month,  and  the  product  will  be  the 
required  interest. 

NOTE,— Since  a  month  is  reckoned  at  30  days,  any  number  of 
days  is  reduced  to  decimals  of  a  month  by  dividing  the  days  by  3. 

EXAMPLES. 

1.  What  is  the  interest  of  $327.50  for  3  years  7  months 
and  13  days,  at  7  per  cent  ? 

OPERATION. 

3yrs.=3Qmos.  $327.50 

7mos.  .07 

13  days— A\mos.  12)22.9250     =int.  for  1  year. 

Timer=43.4jwos.  1.9104  +  =int.  for  1  month. 

NOTE.— The  method  em-  43.4^   =time  in  months, 

ployed,  and  the  number  of  6368 
decimal  places  used,  in  com- 
puting  interest,  may  affect 
the  mills,  and  possibly,  the 

last  figure  in  cents.  It  is  best  7  64 1 6 

to  use  4  places  of  decimals.  $32.97504  Ans. 

2.  What  is  the  interest  of  $1728.60,  at  7  per  cent,  for 

2  years  6  months  and  21  days  ? 

3.  What  is  the  interest  of  $288.30,  at  7  per  cent,  for 

I  year  8  months  and  27  days  ? 

4.  What  is  the  interest  of  $576.60,  at  6  per  cent,  for 
10  months  aucl  18  days? 

5.  What  is  the  interest  of  $854.42,  at  6  per  cent,  for 

3  months  and  9  days  ? 

6.  What  is  the  interest  of  $1153.20,  at  6  per  cent,  for 

I 1  months  and  6  days  ? 

255.  How  do  you  find  the  interest  for  years,  months  and  days  by  the 
second  method  ? 


SIMPLE   INTEREST.  245 

7.  What  is  the  interest  of  $2306.54,  at  5  per  ceut,  for 
7  months  and  28  days  ? 

8.  What  is  the  interest  of  $4272.10,  at  5  per  cent,  for 
10  months  and  28  days? 

9.  What  is  the  interest  of  $1620,  at  4  per  cent,  for  5  years 
and  24  days  ? 

10.  What  is  the  interest  of  $2430.72,  at  4  per  cent,  for 
10  years  and  4  months  ? 

11.  What  is  the  interest  of  $3689.45,  at  7  per  cent,  for 
4  years  and  7  months  ? 

12.  What  is  the  interest  01  $2945.96,  at  7  per  cent,  for 
7  years  and  3  days  ? 

13.  WThat  is  the  interest,  at  8  per  cent,  of  $675.89,  for 
3  years  6  months  and  6  days  ? 

14.  What  is  the  interest,  at  8  per  cent,  on  $12324,  for 

3  years  and  4  months  ? 

15.  What  is  the  interest,  at  9  per  cent,  on  $15328.20,  for 

4  years  and  7  months  ? 

16.  What  is  the  interest  of  $69450  for  1  year  2  months 
and  12  days,  at  9  per  cent  ? 

17.  What  is  the  interest  of  $216.984  for  3  years  5  months 
and  15  days,  at  10  per  cent  ? 

18.  What  is  the  interest  of  $648.54  for  7  years  6  months, 
at  4J  per  cent  ? 

19.  What  is  the  interest  of  $1297.10  for  8  years  5  months, 
at  5 1  per  cent  ? 

20.  What  is  the  interest  of  $864.768  for  9  months  25  days, 
at  6 \  per  cent  ? 

21.  What  is  the  interest  of  $2594.20  for  10  months  and  9 
days,  at  7 1  per  cent? 

22.  What  is  the  amount  of  $2376.84  for  3  years  9  months 
and  12  days,  at  8  J  per  ceut  ? 

23.  What  is  the  amount  of  $5148.40  for  7  years  11  months 
and  23  days,  at  9 J  per  cent  ? 

24.  What  is  the  amount  of  $3565.20  for  3  years  9  months, 
at  10 J  per  cent? 


24:6  SIMPLE   INTEREST. 

25.  What  is  the  amount  of  $125.75  for  1  year  9  months 
and  27  days,  at  7  per  cent  ? 

26.  What  is  the  amount  of  $256  for  10  months  15  days,  at 
7  J  per  cent  ? 

27.  What  is  the  interest  on  a  note  of  $264.42,  given  Janu- 
ary 1st,  1852,  and  due  Oct.  10th,  1855,  at  4  per  cent? 

28.  Gave  a  note  of  $793.26  April  6th,  1850,  on  interest  at 
7  per  cent :  what  is  due  September  10th,  1852  ? 

29.  What  amount  is  due  on  a  note  of  hand  given  June  7th, 
1850,  for  $512.50,  at  6  per  cent,  to  be  paid  Jan.  1st,  1851  ? 

30.  What  is  the  interest  on  $1250.75  for  90  days,  at  10 
per  cent  ? 

31.  What  is  the  amount  of  $71.09  from  Feb.  8th,  1848,  to 
Dec.  7th,  1852,  at  6  j  per  cent  ? 

32.  What  will  be  due  on  a  note  of  $213.27  on  interest 
after  90  days,  at  7  per  cent,  given  May  19th,  1836,  and  pay- 
able October  16th,  1838  ? 

33.  What  is  the  interest  of  $426.54,  from  August  15th, 

1837,  to  March  13th,  1840,  at  7  per  cent? 

34.  What  is  the  interest  of  $2132.70,  from  Nov.  17th, 

1838,  to  Feb.  2d,  1839,  at  7J  per  cent? 

35.  What  is  the  interest  of  $38463,  from  April  27th,  1815, 
to  Sept.  2d,  1824,  at  8  per  cent  ? 

36.  What  is  the  interest  of  $14231.50,  from  June  29th, 
1840,  to  April  30th,  1845,  at  8J  per  cent? 

37.  What  is  the  interest  of  $426.50,  from  Sept.  4th,  1843, 
to  May  4,  1849,  at  9  per  cent? 

38.  What  is  the  interest  of  $4320,  from  Dec.  1st,  1817,  to 
Jan.  22d,  1833,  at  9J  per  cent?" 

39.  What  is  the  amount  of  $397.16,  from  March  24,  1824, 
to  March  31st,  1835,  at  10|  per  cent  ? 

40.  What  is  the  amount  of  $328.12,  from  July  4th,  1809, 
to  Feb.  15th,  1815,  at  3  per  cent  ? 

41.  What  is  the  amount  of  $164.60,  from  Sept.  27th,  1845, 
to  March  24,  1855,  at  1J  per  cent? 

42.  What  is  the  amount  of  $1627.50,  from  July  4th,  1839, 
to  August  1st,  1855,  at  8  per  cent? 


PARTIAL  PAYMENTS.  24:7 


CASE    III. 

256.   When  the  principal  is  in  pounds,  shillings  and 
pence. 

1.  What  is  the  interest,  at  7  per  cent,  of  £27  15s.  9d., 
for  2  years  ? 

OPERATION. 

ANALYSIS.—  The  interest  on  pounds  £27   15s.  9J.  =  27.7875 

and  decimals  of  a  pound  is  found  in  Q>J 
the  same  way  as  the  interest  on  dol- 

lars and  decimals  of  a  dollar:  after  1.945125 

which  the  decimal  part  of  the  interest  2 
may    be    reduced    to    shillings    and 


Ans.  £3  178. 

1.  Reduce  the  shillings  and  pence  to  the  decimal  of  a 
pound  and  annex  the  result  to  the  pounds. 

II.  Find  the  interest  as  though  the  sum  were  United 
States  Money,  after  which  reduce  the  decimal  part  to  shil- 
lings and  pence. 

2.  What  is  the  interest  of  £67  19s.  Qd.}  at  6  per  cent,  for 
3  years  8  months  16  days  ? 

3.  What  is  the  interest  of  £127  15s.  4d.,  at  6  per  cent, 
for  3  years  and  3  months  ? 

4.  What  is  the  interest  of  £107  16s.  IQd.,  at  7  per  cent, 
for  3  years  6  months  and  6  days  ? 

5.  What  will  £279  13s.  8d.  amount  to  in  3  years  and  a 
half,  at  5J  per  cent  per  annum? 

PARTIAL  PAYMENTS. 

257..  A  PARTIAL  PAYMENT  is  a  payment  of  a  part  of  a  note 
or  bond. 

We  shall  give  the  rule  established  in  New  York  (see 
Johnson's  Chancery  Reports,  vol.  I.  page  17),  for  computing 
the  interest  on  a  bond  or  note,  when  partial  payments  have 
been  made.  The  same  rule  is  also  adopted  in  Massachusetts, 
and  in  most  of  the  other  states. 

256.  How  do  you  find  the  interest  when  the  principal  is  in  pounds, 
shillings  and  pence  ? 


248  PARTIAL   PAYMENTS. 

RULE. — I.  Compute  the  interest  on  the  principal  to  the 
time  of  the  first  payment,  and  if  the  payment  exceed  this 
interest,  add  the  interest  to  the  principal  and  from  the  sum 
subtract  the  payment :  the  remainder  forms  a  new  principal : 

II.  But  if  the  payment  is  less  than  the  interest,  take  no 
notice  of  it  until  other  payments  are  made,  which  in  all, 
shall  exceed  the  interest  computed  to  the  time  of  the  last 
payment :  then  add  the  interest,  so  computed,  to  the  princi- 
pal, and  from  the  sum  subtract  the  sum  of  the  payments : 
the  remainder  will  form  a  new  principal  on  which  interest 
is  to  be  computed  as  before. 

NOTE  — In  computing  interest  on  notes,  observe  that  the  day  on 
which  a  note  is  dated  and  the  day  on  which  it  falls  due,  are  not 
both  reckoned  in  determining  the  time,  but  one  of  them  is  always 
excluded.  Thus,  a  note  dated  on  the  first  day  of  May  and  falling 
due  on  the  16th  of  June,  will  bear  interest  but  one  month  and 
1 5  days. 

EXAMPLES. 


$349.998  Buffalo,  May  1st,  1826. 

1.  For  value  received,  I  promise  to  pay  James  Wilson  or 
order,  three  hundred  and  forty-nine  dollars  ninety-nine  cents 
and  eights  mills  with  interest  at  6  per  cent. 

James  Pay  well. 

On  this  note  were  endorsed  the  following  payments  : 
Dec.  25th,  1826  Received  $49.998 
July  10th,  1827  "  $  4.998 
Sept.  1st,  1828  "  $15.008 
June  14th,  1829  "  $99.999 
What  was  due  April  15th,  1830  ? 

Principal  on  int.  from  May  1st,  1826,  -    -    -    -  $349.998 
Interest  to  Dec.  25th,  1826,  time  of  first  pay- 
ment, 7  months  24  days     13.649  + 

Amount,     -    -    -  $363.647 


257.  What  is  a  partial  payment?    What  is  the  rule  for  computing 
Interests  when  there  are  partial  payments  ? 


PARTIAL   PAYMENTS.  249 

Payment  Dec.  25th,  exceeding  interest  then  due  $  49.998 

Remainder  for  a  new  principal $313.649 

Interest  of  $313.649  from  Dec.  25,  1826,  to 

June  14th,  1829,  2  years  5  months  19  days,  -  $  46.4721 

Amount "$360.1211 

Payment,  July  10th,  1827,  .less  than  {*  ^  QQO 

interest  then  due )  *     ' 

Payment,  Sept.  1st,  1828 15.008 

Their  sum  less  than  interest  then  due  -  $20.006 
Payment,  June  14th,  1829  -  -  -  -  99.999 
Their  sum  exceeds  the  interest  then  due-  -  -  $120.005 

Remainder  for  a  new  principal,  June  14,  1829,  $240.1161 
Interest  of  $240.168  from  June  14th,  1829,  to 

April  15th,  1830,  10  months  1  day     -    -     -  $  12.0458 

Total  due,  April  15th,  1830  -     -"$252.1619  + 

$3469.327  New  York,  Feb,  6,  1825. 

2.  For  value  received,  I  promise  to  pay  William  Jenks,  or 
order,  three  thousand  four  hundred  and  sixty-nine  dollars  and 
thirty-two  cents,  with  interest  from  date,  at  6  per  cent. 

Bill  Spendthrift. 

On  this  note  were  endorsed  the  following  payments  : 
May  16th,  1828,  received  $  545.76 
May  16th,  1830,         "       $1276.00 
Feb.    1st,  1831,         "       $2074.72 

What  remained  due  Aug   llth,  1832  ? 

3.  A's  note  of  $635.84  was  dated  September  5,  1817,  on 
which  were   endorsed  the  following  payments,  viz.  :    Nov. 
13th,  1819,  $416.08  ;  May  10th,  1820,  $152.00  :  what  was 
due  March  1st,  1821,  the  interest  being  6  per  cent? 

LEGAL  INTEREST, 

258.  Legal  Interest  is  the  interest  which  the  law  permits 
a  person  to  receive  for  money  which  he  loans,  and  the  laws 
do  not  favor  the  taking  of  a  higher  rate.  In  most  of  the 
States  the  rate  is  fixed  at  6  per  cent ;  in  New  York,  South 
Carolina  and  Georgia,  it  is  7  ;  and  in  some  of  the  States  the 
rate  is  fixed  as  high  as  10  per  cent 


250  PROBLEMS   IN   INTEREST. 

PROBLEMS    IN    INTEREST. 

259.  In  all  questions  of  Interest  there  are  four  things  con- 
sidered, viz. : 

1st,  The  principal  ;  2d,  The  rate  of  interest ;  3d,  The 
time  ;  and  &th,  The  amount  of  interest. 

If  three  of  these  are  known,  the  fourth  can  be  found, 

I.  Knowing,  the  principal,  rate,  and  time,  to  find  the  inter- 
est.    This  case  has  already  been  considered. 

II.  Knowing  the  interest,  time,  and  rate,  to  find  the  prin- 
cipal. 

Cast  the  interest  on  one  dollar  for  the  given  time,  and  then 
divide  the  given  interest  by  it — the  quotient  ivill  be  the  princi- 
pal. 

III.  Knowing  the  interest,  the  principal,  and  the  time,  to 
find  the  rate. 

Cast  the  interest  on  the  principal  for  the  given  time  at  1  per 
cent  and  then  divide  the  given  interest  by  it — the  quotient  will 
be  the  rate  of  interest. 

IV  Knowing  the  principal,  the  interest,  arid  the  rate,  to 
find  the  time. 

Cast  the  interest  on  the  given  principal  at  the  given  rate 
for  1  year  and  then  divide  the  interest  by  it — the  quotient 
will  be  the  time  in  years  and  decimals  of  a  year. 

EXAMPLES 

1.  The  interest  of  a  certain  sum  for  4  years,  at  7  per  cent, 
is  $266  :  what  is  the  principal? 

2.  The  interest  of  $3675,  for  3  years,  is  $171.15  :  what  is 
the  rate? 

3.  The  principal  is  $459,  the  interest  $183.60,  and  the 
rate  8  per  cent :  what  is  the  time  ? 

4.  The  interest  of  a  certain  sum,  for  3  years,  at  6  per  cent, 
is  $40.50  :  what  is  the  principal  ? 

5.  The  principal  is  $918,  the  interest  $269.28,  and  the 
rate  4  per  cent :  what  is  the  time  ? 

258.  What  is  legal  interest  ? 

259.  How  many  things  are  considered  in  every  question  of  interest? 
What  arc  they  ?     What  is  the  rule  for  each  ? 


COMPOUND    INTEKEST.  251 


COMPOUND  INTEREST. 

260.  Compound  Interest  is  when  the  interest  on  a  princi- 
pal, computed  to  a  given  time,  is  added  to  the  principal,  and 
the  interest  then  computed  on  this  amount,  as  on  a  new 
principal.  Hence, 

Compute  the  interest  to  the  time  at  which  it  becomes  due  ; 
then  add  it  to  the  principal  and  compute  the  interest  on  the 
amount  as  on  a  new  principal:  add  the  interest  again  to 
the  principal  and  compute  the  interest  as  before  ;  do  the 
same  for  all  the  times  at  which  payments  of  interest  become 
due ;  from  the  last  result  subtract  the  principal,  and  the 
remainder  will  be  the  compound  interest. 

EXAMPLES. 

1.  What  will  be  the  compound  interest,  at  7  per  cent,  of 
$3150  for  2  years,  the  interest  being  added  yearly? 

*  OPERATION. 

$3750.000  principal  for  1st  year. 

$3750  x. 07=     262.500  interest    for  1st  year 

4012.500  principal  for  2d     " 

$4012.50  x. 07=     280.875  interest    for  2d     " 

4293.375     amount  at  2  years. 
1st  principal  3750.000 
Amount  of  interest  $543.375. 

2.  If  the  interest  be  computed  annually,  what  will  be  the 
compound  interest  on  $100  for  3  years,  at  6  per  cent? 

3.  What  will  be  the  compound  interest  on  $295.37,  at  6 
per  cent,  for  2  years,  the  interest  being  added  annually  ? 

4.  What  will  be  the  compound  interest,  at  5  per  cent,  of 
$1875,  for  4  years? 

5.  What  is  the  amount  at  compound  interest  of  $250,  for 
2  years,  at  8  per  cent  ? 

6.  What  is  the  compound  interest  of  $939.64,  for  3  years, 
at  7  per  cent  ? 

7.  What  will  $125.50  amount  to  in  10  years,  at  4  per  cent 
compound  interest  ? 

260.  What  of  compound  interest  ?    How  do  you  compute  it  ? 


252 


COMPOUND   INTEREST. 


NOTE. — The  operation  is  rendered  much  shorter  and  easier,  by 
taking  the  amount  of  1  dollar  for  any  time  and  rate  given  in  the 
following  table,  and  multiplying  it  by  the  given  principal ;  the 
product  will  be  the  required  amount,  from  which  subtract  the 
given  principal,  and  the  result  will  be  the  compound  interest.* 

TABLE. 

Which  shows  the  amount  of  $1  or  £1,  compound  interest,  from  1  year 
to  20,  aud  at  the  rate  of  3,  4,  5,  6,  and  7  per  cent. 


Years.  jiSper  cent. 

4  per  rent.io  per  cent. 

ti  per  cent. 

'  per  ci-Bt. 

V«ars. 

1 

1.03000 

1.04000 

1.05000 

1.06000 

1.07000 

1 

2 

1.0(5090 

1.08160 

1.10250 

1.12360 

1.14490 

2 

3 

1.09272 

1.12486 

1.15762 

1.19101 

1.22504       3 

1.135501.109851.21550 

1.26247 

1.31079       4 

5 

1.15927 

1.216'>5 

1.27628 

1.  33822 

1.40255 

5 

6 

1  19405 

1.26531 

1.34009 

1.41851 

1.50073 

6 

7 

1.22987 

1.31593  1.  40710 

1.50363 

1.60578 

7 

8    ft.26677 
9     1.30477 

1.36856!  1.47745 
1.4233111.55132 

1.59384 
1.C8947 

1.71818 
1.83845 

8 
9 

10     1.34391 

1.480:38 

1.62889 

1.79084 

1.96715 

10 

11     11.38433 

1.53945 

1.71033 

1.89829 

2.10485 

11 

12 

1.4257(5 

1.60103 

1.79585 

2.012192.25219 

12 

13 

1.46853 

1.66507 

1.88564 

2.13292240984 

13 

14 

1.5!  258 

1.73167 

1.97993 

2.260902,57853 

14 

15 

1.55796 

1.80094 

2.07892 

2.396552.75903 

15 

16 

1.60470 

1.8729812.18287 

2.54035  2.95216 

16 

17 

1.  (55284 

1.94790J2.  29201 

2.  69277  ;  3.  15881 

17 

18 

1.70243 

2.02581 

2.40661 

2.854333.37993 

18 

19 

1.75350 

2.10684 

2.52695 

3.025593.61652 

19 

20 

1.80611 

2.19112 

,2.  (55329  3.  2071  3  i  3.  86968  1     20 

NOTE. — When  there  are  months  and  days  in  the  time,  find  the 
amount  for  the  years,  and  on  this  amount  cast  the  interest  for  the 
mcnths  and  days :  this,  added  to  the  last  amount,  will  be  the  re- 
quired amount  for  the  whole  time. 

8.  What  is  the  amount  of  $96.50  for  8  years  and  6  months, 
interest  being  compounded  annually  at  7  per  cent  ? 

9.  What  is  the  compound  interest  of  $300  for  5  years 
8  months  and  15  days,  at  6  per  cent  ? 

10.  What  is  the  compound  interest  of  $1250  for  3  years 
3  months  and  24  days,  at  7  per  cent  ? 

11.  What  will  $56.50  amount  to  in  20  years  and  4  months, 
at  5  per  .cent  compound  interest  ? 

*  The  result  may  differ  in  the  mills  place  from  that  obtained  by  the 
other  rule. 


DISCOUNT.  253 


DISCOUNT.     x 

261.  DISCOUNT  is  an  allowance  made  for  the  payment  of 
money  before  it  is  due. 

THE  FACE  of  a  note  is  the  amount  named  in  the  note.* 

NOTE. — DAYS  OP  GRACE  are  days  allowed  for  the  payment  of 
a  note  after  the  expiration  of  the  time  named  on  its  face.  By 
mercantile  usage  a  note  does  not  legally  fall  due  until  3  days 
after  the  expiration  of  the  time  named  on  its  face,  unless  the  note 
specifies  without  grace. 

Days  of  grace,  however,  are  generally  confined  to  mercantile 
paper  and  to  notes  discounted  at  banks. 

262.  The  PRESENT  VALUE  of  a  note  is  such  a  sum  as  being 
put  at  interest  until  the  note  becomes  due,  would  increase  to 
an  amount  equal  to  the  face  of  the  note. 

The  discount  on  a  note  is  the  difference  between  the  face 
of  the  note  and  its  present  value. 

1.  I  give  my  note  to  Mr.  Wilson  for  $10 7,  payable  in 
1  year  :  what  is  the  present  value  of  the  note  if  the  interest 
is  7  per  cent.  ?  what  the  discount  ? 

OPERATION. 

ANALYSIS.— Since  1  dollar  in  1  year  $107 -f- 1,07— $100. 
at  7  per  cent,  will  amount  to  $1.07,  the  PROOF 

present  value   will   be  as  many  dollars   yn4.   (frinn   1,1. <6      *r 

as  $1.07  is  contained  times   in  the  face  t,  .   \,      ^        \nA 
of   the  note:    viz.,  $100:   and  the  dis-  -Principal, 
count  will  be  $107- $100= $7:  hence,         Amount,  $107 

Discount,  7 

Divide  the  face  of  the  note  by  1  dollar  plus  the  interest  of 
1  dollar  for  the  given  time,  and  the  quotient  will  be  the  pre- 
sent value :  take  this  sum  from  the  face  of  the  note  and  the 
remainder  will  be  the  discount. 


261.  What  is  discount  ?    What  is  the  face  of  a  note  ?    What  are  days 
of  grace? 

362.   What  is  present  value  ?    What  is  the  discount  ?    How  do  you 
find  the  present  value  of  a  note  ? 

*  See  Appendix,  page  3l(X 


254  DISCOUNT. 

EXAMPLES. 

1.  What  is  the  present  value  of  a  note  for  $1828,75,  eke 
in  1  year,  and  bearing  an  interest  of  4  J  per  cent  ? 

2.  A  note  of  $1651.50  is  due  in  11  months,  but  the  person 
to  whom  it  is  payable  sells  it  with  the  discount  off  at  6  per 
cent :  how  much  shall  he  receive  ? 

NOTE.— When  payments  are  to  be  made  at  different  times,  find 
the  present  value  of  the  sums  separately,  and  tfieir  sum  will  be  the 
present  value  of  the  note. 

3  What  is  the  present  value  of  a  note  for  $10500,  on  which 
$900  are  to  be  paid  in  6  mouths  ;  $2700  in  one  year  ;  $3900 
in  eighteen  months  ;  and  the  residue  at  the  expiration  of  two 
years,  the  rate  of  interest  being  6  per  cent  per  annum  ? 

4.  What  is  the  discount  of  <£4500,  one-half  payable  in  six 
months  and  the  other  half  at  the  expiration  of  a  year,  at  7 
per  cent  per  annum  ? 

5.  What  is  the  present  value  of  $5760,  one-half  payable  in 

3  months,  one-third  in  6  months,  and  the  rest  in  9  months, 
at  6  per  cent  per  annum  ? 

6.  Mr.  A  gives  his  note  to  B  for  $720,  one-half  payable  in 

4  months  and  the  other  half  in  8  months  ;  what  is  the  present 
value  of  said  note,  discount  at  5  per  cent  per  annum  ? 

7.  What  is  the  difference  between  the  interest  and  discount 
of  $750,  due  nine  months  hence,  at  7  per  cent  ? 

8.  What  is  the  present  value  of  $4000  payable  in  9  months, 
discount  4J  per  cent  per  ami  am  ? 

9.  Mr.   Johnson   has   a  note   against    Mr.    Williams  for 
$2146.50,  dated  August  17th,  1838,  which  becomes  due  Jan. 
llth,  1839  :  if  the  note  is  discounted  at  6  per  cent,  what 
ready  money  must  be  paid  for  it  September  25th,  1838  ? 

10.  C  owes  D  $3456,  to  be  paid  October  27th,  1842  ;  C 
wishes  to  pay  on  the  24th  of  August,  1838,  to  which  D  con- 
sents ;  how  much  ought  D  to  receive,  interest  at  6  per  cent  ? 

11.  What  is  the  present  value  of  a  note  of  $4800,  due  4 
years  hence,  the  interest  being  computed  at  5  per  cent  per 
annum  ? 

12.  A  man  having  a  horse  for  sale,  offered  it  for  $225  cash 
in  hand,  or  $230  at  9  months  ;  the  buyer  chose  the  latter  : 
did  the  seller  lose  or  make  by  his  offer,  supposing  money  to 
be  worth  7  per  cent  ? 


BANK  DISCOUNT,  255 


BANK  DISCOUNT. 

263.  BANK  DISCOUNT  is  the  charge  made  by  a  bank  for  the 
payment  of  money  on  a  note  before  it  becomes  due. 

By  the  custom  of  banks,  this  discount  is  the  interest  on  the 
amount  named  in  a  note,  calculated  from  the  time  the  note 
is  discounted  to  the  time  when  it  falls  due  ;  in  which  time 
the  three  days  of  grace  are  always  included. 

The  interest  is  always  paid  in  advance. 

RULE  — Add  3  days  to  the  time  which  the  note  has  to  run, 
and  then  calculate  the  interest  for  that  time  at  the  given  rate. 

EXAMPLES. 

1.  What  is  the  Dank  discount  of  a  note  for  $350,  payable 
3  months  after  date,  at  7  per  cent  interest  ? 

2.  What  is  the  bank  discount  of  a  note  of  $1000  payable 
in  60  days,  at  6  per  cent  interest  ? 

3.  A  merchant  sold  a  cargo  of  cotton  for  $15720,  for  which 
he  receives  a  note  at  6  months  :  how  much  money  will  he 
receive  at  a  bank  for  this  note,  discounting  it  at  6  per  cent 
interest  ? 

4.  What  is  the  bank  discount  on  a  note  of  $556. 2 1  paya- 
ble in  60  days,  discounted  at  6  per  cent  interest? 

5.  A  has  a  note  against  B  for  $3456,  payable  in  three 
months  ;  he  gets  it  discounted  at  7  per  cent  interest :  how 
much  does  he  receive  ? 

6.  What  is  the  bank  discount  on  a  note  of  367.47,  having 
1  year,  1  month,  and  13  days  to  run,  as  shown  by  the  face  of 
the  note,  discounted  at  7  per  cent  ? 

7-  For  value  received,  I  promise  to  pay  to  John  Jones,  on 
the  20th  of  November  next,  six  thousand  five  hundred  and 
seventy-nine  dollars  and  15  cents.  What  will  be  the  discount 
on  this,  if  discounted  on  the  1st  of  August,  at  6  per  cent  per 
annum  ? 

263.  What  is  bank  discount  ?  How  is  interest  calculated  by  the 
custom  of  banks  ?  How  is  the  interest  paid  ?  How  do  you  find  the 
interest  ? 


256  BANK  DISCOUNT. 

8.  A  merchant  bought  115  barrels  of  flour  at  $7.50  cents 
a  barrel,  and  sells  it  immediately  for  $9.75  a  barrel,  for 
which  he  receives  a  good  note,  payable  in  6  months.  If  he 
should  get  this  note  discounted  at  a  bank,  at  6  per  cent,  what 
will  be  his  gain  on  the  flour  ? 

264.  To  make  a  note  due  at  a  future  lime,  whose  present 
value  shall  be  a  given  amount. 

1.  For  what  sum  must  a  note  be  drawn  at  3  months,  so 
that  when  discounted  at  a  bank,  at  6  per  cent,  the  amount 
received  shall  be  $500  ? 

ANALYSIS — If  we  find  the  interest  on  1  dollar  for  the  given 
time,  and  then  subtract  that  interest  from  1  dollar,  the  remainder 
will  be  the  present  value  of  1  dollar,  due  at  the  expiration  of  that 
time.  Then,  the  number  of  times  which  the  present  value  of 
the  note  contains  the  present  value  of  1  dollar,  will  be  the  num- 
ber of  dollars  for  which  the  note  must  be  drawn :  hence, 

Divide  the  present  value  of  the  note  by  the  present  value  of 
1  dollar,  reckoned  for  the  same  time  and  at  the  same  rate  of 
interest ,  and  the  quotient  will  be  the  face  of  the  note, 

OPERATION. 

Interest  of  $1  for  the  time,  3mo.  and  Ma. =$0.0155,  which 
taken  from  $1,  gives  present  value  of  $1=0.9845;  then,  $500^- 
0.9845= $507.872-1-  =face  of  note. 

PROOF. 

Bank  interest  on  $507.872  for  3  months,  including  3  days  of 
grace,  at  6  per  cent =7.872,  which  being  taken  from  the  face  of 
the  note,  leaves  $500  for  its  present  value, 

EXAMPLES, 

1 .  For  what  sum  must  a  note  be  drawn,  at  7  per  cent, 
payable  on  its  face  in  1  year  6  months  and  1 5  days,  so  that 
when  discounted  at  bank  it  shall  produce  $307.27  ? 

2.  A  note  is  to  be  drawn  having  on  its  face  8  months  and 
1 2  days  to  run,  and  to  bear  an  interest  of  7  per  cent,  so  that 
it  will  pay  a  debt  of  $5450  :  what  is  the  amount  ? 

364.  How  do  you  make  a  note  payable  at  a  future  time,  whose  pre- 
sent value  shall  be  a  given  amount  ? 


EQUATION   OF   PAYMENTS.  257 

3.  What  sum,  6  months  and  9  days  from  July  18th,  1856, 
drawing  an  interest  of  6  per  cent,  will  pay  a  debt  of  $674.89 
at  bank,  on  the  1st  of  August,  1856  ? 

4.  Mr  Johnson  has  Mr.  Squires'  note  for  $814.57,  having 
4  months  to  run,  from  July  13th,  without  interest.     On  the 
first  of  October  he  wishes  to  pay  a  debt  at  bank  of  $750.25, 
and  discounts  the  note  at  5  'per  cent  in  payment :  how  much 
must  he  receive  back  from  the  bank  ? 

5.  Mr.  Jones,  on  the  1st  of  June,  desires  to  pay  a  debt  at 
bank  by  a  note  dated  May  1 6th,  having  6  months  to  run  and 
drawing  7  per  cent  interest :  for  what  amount  must  the  note 
be  drawn,  the  debt  being  $1683.75  ? 

6  Mr.  Wilson  is  indebted  at  the  bank  in  the  sum  of 
$367.464,  which  he  wishes  to  pay  by  a  note  at  4  months 
with  interest  at  7  per  cent :  for  what  amount  must  the  note 
be  drawn  ? 

EQUATION  OF  PAYMENTS, 

265.  EQUATION  OF  PAYMENTS  is  the  operation  of  finding  the 
mean  time  of  payment  of  several  sums  due  at  different  times, 
so  that  no  interest  shall  be  lost  or  gained.* 

1.  If  I  owe  Mr.  Wilson  2  dollars  to  be  paid  in  6  months, 
3  dollars  to  be  paid  in  8  months,  and  1  dollar  to  be  paid  in 
12  months,  what  is  the  mean  time  of  payment  ? 

OPERATION. 

Int.  of  $2  for    6rao.=int.  of  $1  for  12mo.  2x    6—12 

"     of  $3  for    8rao;— int.  of  $1  for  24??io.  3x    8  =  24 

"     of  $1  for  12wio.=mt.  of  $1  for  12 mo.  I  x  12^12 

$6  48  48 

ANALYSIS. — The  interest  on  all  the  sums,  to  the  times  of  pay- 
ment,  is  equal  to  the  interest  of  $1  for  48  months.  But  48  is 
equal  to  the  sum  of  all  the  products  which  arise  from  multiplying 
each  sum  by  the  time  at  which  it  becomes  duo:  hence,  the  sum 
of  the  products  is  equal  to  the  time  which  would  be  necessary  for 
$1  to  produce  the  game  interest  as  would  be  produced  by  all  the 
principals. 

*  The  mean  time  of  payment  is  sometimes  found  by  first  finding  the 
jyrcsent  value  of  each  payment ;  but  the  rule  here  given  has  the  sanc- 
tion of  the  best  authorities  in  this  country  and  England. 
17 


253  EQUATION   OF  PAYMENTS. 

'  $1  will  produce  a  certain  interest  in  48  months,  in  what  time 
will  $6  (or  the  sum  of  the  payments)  produce  the  same  interest  ? 
The  time  is  obviously  found  by  dividing  48  (the  sum  of  the  pro- 
ducts) by  $6,  (the  sum  of  the  payments.) 
Hence,  to  find  the  mean  time, 

Multiply  each  payment  by  the  time  before  it  becomes  due, 
and  divide  the  sum  of  the  products  by  the  sum  of  the  pay- 
ments :  the  quotient  will  be  the  mean  time. 

EXAMPLES. 

1.  B.  owes  A  $600  ;  $200  is  to  be  paid  in  two  months, 
$200  in  four  months,  and  $200  in  six  months  :  what  is  the 
mean  time  for  the  payment  of  the  whole  ? 

OPERATION. 
200x2-=  400 

ANALYSIS.  —  We  here   multiply  each     200x4—   800 
sum  by  the  time  at  which  it  becomes     QHA      f_ionn 
due,  and  divide  the  sum  of  the  products     JUU 
by  the  sum  of  the  payments.  6|00        )24|00 

Ans.  4  months. 

2.  A  merchant  owes  $600,  of  which  $100  is  to  be  paid  in 
4  months,  $200  in   10  months,   and  the   remainder  in   16 
months  :  if  he  pays  the  whole  at  once,  in  what  time  must  he 
make  the  payment  ? 

3.  A  merchant  owes  $600  to  be  paid  in  12  months,  $800 
to  be  paid  in  6  months,  and  $900  to  be  paid  in  9  months  : 
what  is  the  equated  time  of  payment  ? 

4.  A  owes  B  $600  ;  one-third  is  to  be  paid  in  6  months, 
one-fourth  in  8  months,  and  the  remainder  in  12  months  : 
what  is  the  .mean  time  of  payment  ? 

5.  A  merchant  has  due  him  $300  to  be  paid  in  60  days, 
$500  to  be  paid  in  120  days,  and  $750  to  be  paid  in  180 
days  :   what  is  the  equated  time  for  the  payment  of  the 


6.  A  merchant  has  due  him  $1500  :  one-sixth  is  to  bo 
paid  in  2  months,  one-third  in  3  months,  and  the  rest  in  6 
months  :  what  is  the  equated  time  for  the  payment  of  the 
whole  ? 

265.  What  is  equation  of  payments  ?    How  do  you  find  the  mean  or 
equated  time  ? 


EQUATION   OF   PAYMENTS.  259 

7.  I  owe  $1000  to  be  paid  on  the  first 'of  January,  $1500 
on  the  1st  of  February,  $3000  on  the  1st  of  March,  and 
$4000  on  the  15th  of  April :  reckoning  from  the  1st  of  Janu- 
ary, and  calling   February  28  days,  on  what  day  must  the 
money  be  paid  ? 

NOTE. — If  one  of  the  payments,  as  in  the  above  example,  is  due 
on  the  day  from  which  the  equated  time  is  reckoned,  its  corres- 
ponding product  will  be  notliing,  but  the  payment  must  still  be 
added  in  finding  the  sum  of  the  payments, 

8.  I  owe  Mr  Wilson  $100  to  be  paid  on  the  15th  of  July, 
$200  on  the  15th  of  August,  and  300  on  the  9th  of  Septem- 
ber :  what  is  the  mean  time  of  payment  ? 

OPERATION 

From  1st  of  July  to  1st  payment  14  days 

"        "         "      to  2d  payment  45  days. 

"      to  3d  payment  70  days. 

100x14=   1400 
200x45=  9000 

Tlien  by  rule  given  above  we     300  X  70  =  2 1 000 
have, 


600     6|00)314|00 

fili  • 

Hence,  the  equated  time  is  52^  days  from  the  1st  of  July  ;  that 
is,  on  the  22d  day  of  August. 

But  if  we  estimate  the  time  from  the  15th  of  July  we  shall  have 

From  July  15th  to  1st  payment    0  days. 
"         "  to  2d  payment  30  days. 

"         "  to  3d  payment  54  days. 

Then,  100  x    0=     000 

200x30=  COOO 
300x54  =  16200 
600 


Hence,  the  payment  is  due  in  37  days  from  July  15th;  or,  on 
the  22d  of  August  —  the  same  as  before. 

Therefore  :  Any  day  may  be  taken  as  the  one  from,  which 
the  mean  time  is  reckoned. 

NOTE.  —  If  one  payment  is  due  on  the  day  from  which  the  time  is 
reckoned,  how  do  you  treat  it  ?  Can  you  compute  the  time  from  any 
day? 


260  ASSESSING    TAXES. 

9.  Mr.  Jones  purchased  of  Mr.  Wilson,  on  a  credit  of  six 
months,  goods  to  the  following  amounts  : 

15th  of  January,    a  bill  of  $3t50, 

10th  of  February,  a  bill  of  3000, 
6th  of  March,  a  bill  of  2400, 
8th  of  June,  a  bill  of  2250. 

He  wishes,  on  the  1st  of  July,  to  give  his  note  for  the 
amount :  at  what  time  must  it  be  made  payable  ? 

10.  Mr  Gilbert  bought  $4000  worth  of  goods  ;  he  was  to 
pay  $1600  in  five  months,  $1200  in  six  months,  and  the  re- 
mainder in  eight  months  :  what  will  be  the  time  of  credit,  if 
he  pays  the  whole  amount  at  a  single  payment  ? 

11.  A  merchant  bought  several  lots  of  goods,  as  follows  : 

A  bill  of  $650,  June  6th, 
A  bill  of  890,  July  8th, 
A  bill  of  7940,  August  1st. 

Now,  if  the  credit  is  6  months,  how  many  days  from  De- 
cember 6th  before  the  note  becomes  due  ?  At  what  time  ? 

ASSESSING    TAXES. 

26G.  A  tax  is  a  certain  sum  required  to  be  paid  by  the 
inhabitants  of  a  town,  county,  or  state,  for  the  support  of 
government  or  some  public  object.  It  is  generally  collected 
from  each  individual,  in  proportion  to  the  amount  of  his 
property. 

In  some  states,  however,  every  white  male  citizen  over  the 
age  of  twenty-one  years  is  required  to  pay  a  certain  tax. 
This  tax  is  called  a  poll-tax  ;  and  each  person  so  taxed  is 
called  a  poll. 

267.  In  assessing  taxes,  the  first  thing  to  be  done  is  to  make 
a  complete  inventory  of  all  the  property  in  the  town  on  which 
the  tax  is  to  be  laid.  If  there  is  a  poll-tax,  make  a  full  list 
of  the  polls  and  multiply  the  number  by  the  tax  on  each 
poll,  and  subtract  the  product  from  the  whole  tax  to  be 

266.  What  is  a  tax  ?  llow  is  it  generally  collected  ?  What  is  a 
poll-tax  ? 


ASSESSING   TAXES.  2C1 

raised  by  the  town  :  the  remainder  will  be  the  amount  to 
be  raised  on  the  property  Having  done  this,  divide  the 
whole  tax  to  be  raised  by  the  amount  of  taxable  properly 
and  the  quotient  will  be  the  tax  on  $1.  Then  multiply  this 
quotient  by  the  inventory  of  each  individual,  and  the  product 
will  be  the  tax  on  his  property 

EXAMPLES. 

1.  A  certain  town  is  to  be  taxed  $4280  ;  the  property  on 
which  the  tax  is  to  be  levied  is  valued  at  $1000000.  Now 
there  are  200  polls,  each  taxed  $1.40.  The  property  of  A 
is  valued  at  $2800,  and  he  pays  4  polls. 

B's  at  $2400,  pays  4  polls.     E's  at  $7242,  pays  4  polls. 
C's  at  $2530,  pays  2     "         F's  at  $1651,  pays  6     " 
D's  at  $2250,  pays  6     "        G's  at  $1600.80  pays  4  " 

What  will  be  the  tax  on  1  dollar,  and  what  will  be  A's 
tax,  and  also  that  of  each  on  the  list  ? 

First;     $1.40  x  200  =  $280  amount  of  poll-tax. 
$4280— $280  —  4000  amount  to  be  levied  on  property. 
Then,     $4000-i-$1000000=4  mills  on  $1. 
Now,  to  find  the  tax  of  each,  as  A's,  for  example, 

A's  inventory $2800 

_^004 
TT200 
4  polls  at  $1,40  each     -    -       5.60 

A's  whole  tax  -     -    -    -    -  $16.800> 
In  the  same  manner  the  tax  of  each  person  in  the  town- 
ship may  be  found. 

Having  found  the  per  cent,  or  the  amount  to  be  raised  on 
each  dollar,  form  a  table  showing  the  amount  which  certain 
sums  would  produce  at  the  same  rate  per  cent.  Thus,  after 
having  found,  as  in  the  last  example,  that  4  mills  are  to  be 
raised  on  every  dollar,  we  can,  by  multiplying  in  succession 
by  the  numbers  1,  2,  3,  4.  5,  6,  7,  8,  &c.,  form  the  following 

267.  What  is  the  first  thing  to  be  done  in  assessing  a  tax  ?  If  there 
is  a  poll-tax,  how  do  you  find  the  amount  ?  Howxthen  do  you  find  the 
per  cent  of  tax  to  be  levied  on  a  dollar  ?  How  do  you  then  find  the 
amount  to  be  levied  on  each  individual  ? 


262 


ASSESSING   TAXES. 


TABLE 


$      $ 

$            $ 

$               $ 

1  gives  0.004 

20  gives  0  080 

300  gives    1.200 

2     "     0.008 

30     "     OJ20 

400     "       1.600 

3     "     0-012 

40     (<     0.160 

500     "       2.000 

4     "     0.016 

50     "     0.200 

600     "       2.400 

5     "     0.020 

60     "     0.240 

700     "       2.800 

6     "     0.024 

70     "     0.280 

800     "       3.200 

7     "     0.028 

80     "     0.320 

900     "       3.600 

8     "     0.032 

90     "     0.360 

1000     "       4.000 

9     "     0.036 

100     "     0.400 

2000     "       8.000 

10     "     0.040 

200     "     0.800 

3000     "     12.000 

This  table  shows  the  amount  to  be  raised  on  each  sum  in 
the  columns  under  $'s. 

EXAMPLES. 

1.  Find  the  amount  of  B's  tax  from  this  table. 

B's  tax  on  $2000  -     -     is     -     $8.000 

B's  tax  on      400  -    -     is,    -     $1.600 

B's  tax  on  4  polls,  at  $1.40    -     $5  600 

B's  total  tax  -     is     -  $15.200 

2.  Find  the  amount  of  C's  tax  from  the  table. 

C's  tax  on  $2000  -  -  is  -  $8.000 
C's  tax  on  500  -  -  is  -  $2.000 
C's  tax  on  30  -  -  is  -  $0.120 
C's  tax  on  2  polls  -  -  is  -  $2.800 
C's  total  tax  -  -  is  -"$12.920 

In  a  similar  manner,  we  might  find  the  taxes  to  be  paid 
by  D,  E,  &c. 

3.  If  the  people  of  a  town  vote  to  tax  themselves  $1500, 
to  build  a  public  hall,  and  the  property  of  the  town  is  valued 
at  $300.000,  what  is  D's  tax,  whose  property  is  valued  at 
$2450? 

4.  In  a  school  district  a  school  is  supported  by  a  tax  on 
the  property  of  the  district  valued  at  $121340.     A  teacher  is 
employed  for  5  months  at  $40  a  month,  and  contingent  ex- 
penses are  $42,68  ;  what  will  be  a  farmer's  tax  whose  property 
is  valued  at  $3125? 


COINS   AND   CURRENCY.  263 


COINS    AND    CURRENCY. 

268.  Coins  are  pieces  of  metal,  of  gold,  silver,  or  copper,  of 
fixed  values,  and  impressed  with  a  public  stamp  prescribed 
by  the  country  where  they  are  made.  These  are  called 
specie,  and  are  declared  to  be  a  legal  tender  in  payment  of 
debts. 

2(51).  Currency  is  what  passes  for  money.  In  our  country 
there  are  four  kinds. 

1st.  The  coins  of  the  country  : 

M.  Foreign  coins,  having  "a  fixed  value  established  by 
law  : 

3e?.  Bank  notes,  redeemable  in  specie. 

4th.  Paper  money  declared  a  legal  tender,  by  act  of 
Congress. 

NOTE. — The  foreign  coins  most  in  use  in  this  country  are  the 
English  shilling,  valued  at  22  cents  2  mills ;  the  English  sove- 
reign, valued  at  $4,84  ;  the  French  franc,  valued  at  18  cents  6 
mills  ;  and*  the  five-franc  piece,  valued  at  $0.93. 

Although  the  currency  of  the  United  States  is  in  dollars, 
cents  and  mills,  yet  in  some  of  the  States  accounts  are  still 
kept  in  pounds,  shillings  and  pence. 

In  all  the  States  the  shilling  is  reckoned  at  12  pence,  the 
pound  at  20  shillings,  and  the  dollar  at  100  cents. 

The  following  table  shows  the  number  of  shillings  in  a  dol- 
lar, the  value  of  £1  in  dollars,  and  the  value  of  $1  in  the 
fraction  of  a  pound  ? 


In  English  currency, 

4s.  bd.  -  £1=$4.84: 

,  and$l=£T.-i-T. 

In  N.  E.,  Ya  ,  Ky.,  ( 

C                ^1  —  &31 

,  , 

Tenn.,                   j 

$   5, 

an      *          ^TtT- 

In  N.  Y.,  Ohio,  N.  [ 

Carolina-,                j 

8s.        -  £l=$22, 

and  $1—  £  |. 

In  N.  J.,  Pa.,  Del.,  [ 
Md.,                      ) 

Ts.  &d.  -  J61=$2|, 

and$l=£  f 

In  S.  Carolina  &Ga. 

4s.  Sd.  -  «£l:=$4f, 

and  $!=.£,&. 

In  Canada  &  Nova  ) 
Scotia,                   j 

5.,        -  £1=*, 

and  $!=:£  l. 

368.  What  arc  coins? 

V/hat  arc   they  called  ? 

Wliat  is   made  " 

legal  tender? 


26 tt  REDUCTION   OF   CURRENCIES. 


REDUCTION  OF  CURRENCIES. 

270.  Reduction  of  Currencies  is  changing  their  denomina- 
tions without  changing  their  values. 

There  are  two  cases  of  the  Reduction  of  Currencies  : 

1st.  To  change  a  currency  in  pounds,  shillings  and  pence, 

to  United  States  currency. 

2d.  To  change  United  States  currency  to  pounds,  shillings 

and  pence. 

271.  To  reduce  pounds,  shillings  and  pence  to   United 
States  currency. 

1.  What  is  the  value  of  <£3  12s.  Qd.,  New  England  cur- 
rency, in  United  States  money. 

OPERATION. 

ANALYSIS.— Since  £l  =  $3i  the  «£3  12s.  Qd.=£3.G%5 

number  of  dollars  in  £3  12s.  Gd.=  rlnllc   in  ^1    -  31 

£3.625,  will   be   equal   to  £3.625 

taken  3^  times  :  that  is,  to  $12,08 :  1.2084" 

hence,  10.875 

Ans.  $12.083  + 

Multiply  the  amount  reduced  to  pounds  and  the  decimals  of 
a  pound  by  the  number  of  dollars  in  a  pound,  and  the  product 
will  be  the  answer. 

272.  To  reduce  United  States  money  to  pounds,  shillings 
and  pence. 

1.  What  is  the  value  of  $375.81,  in  pounds,  shillings  and 
pence,  New  York  currency  ? 

ANALYSIS.  —  Since    $!=££,  the 

number  of  pounds  in  $375.87  will  bo  OPERATION. 

equal  to  this  number  taken  £  times  :  $375.87  X  -?  =<£150  348 

that  is,  equal  to  £150.348=£150  6s.  =^E150  6s    Hid 

. :  hence, 


200.  What  is  currency  ?  How  many  kinds  arc  there  ?  What  foreign 
coins  are  most  used  in  this  country?  What  are  the  denominations  of 
United  States  currency  ?  What  denominations  are  sometimes  used  in 
the  States  ? 

270.  What  is  reduction  of  currencies  ?    How  many  kinds  of  reduc- 
tion arc  there  ?    What  arc  they  ? 

271.  What  is  the  rule  for  reducing  from  pounds,  shillings  rind  pence 
to  United  States  money  ? 


EXCHANGE.  265 

Multiply  the  amount  by  that  fraction  of  a  pound  which 
denotes  the  value  q/1  $1,  and  the  product  will  be  the  answer  in 
pounds  and  decimals  of  a  pound. 

EXAMPLES 

1.  What  is  the  value  of  £127   18s.  6d.,  New  England 
currency,  in  United  States  money  ? 

2.  What  is  the  value  of  $2863.75  in  pounds,  shillings  and 
pence,  Pennsylvania  currency  ? 

3.  What  is  the  value  of  £459  3s.  Qd.,  Georgia  currency,  in 
United  States  money  ? 

4.  What  is  the  value  of  $973.28  in  pounds,  shillings  and 
pence,  North  Carolina  currency  ? 

5.  What  is  the  value  in  United  States  money  of  £637  18s. 
8d.,  Canada  currency  ? 

6.  Reduce   $102.85  to  English  money  ;    to  Canada  cur- 
rency ;  to  New  England  currency  ;  to  New  York  currency  ; 
to  Pennsylvania  currency  ;  to  South  Carolina  currency. 

7.  Reduce  £51  13s.  OJtf.  English  money  ;  £62  10s.  Can- 
ada currency  ;    £75  New  England  currency  ;    £100   New 
York  currency  ;  £193  15s.  Pennsylvania  currency  ;  and  £58 
6s.  7Jrf.  Georgia  currency,  to  United  States  money. 

EXCHANGE. 

273.  EXCHANGE  denotes  the  payment  of  a  sum  of  money 
by  a  person  residing  in  one  place  to  a  person  residing  in  an- 
other. The  payment  is  usually  made  by  means  of  a  bill  of 
exchange. 

A  BILL  OF  EXCHANGE  is  an  order  from  one  person  to  another 
directing  the  payment  to  a  third  person  named  therein  of  a 
certain  sum  of  money  : 

1.  He  who  writes  the  open  letter  of  request  is  called  the 
drawer  or  maker  of  the  bill. 

2.  The  person  to  whom  it  is  directed  is  called  the  draw'ee. 


272.  What  is  the  rule  for  reducing  from  United  States  money  to 
pounds,  shillings  and  pence  ? 

273.  What  does  exchange  denote  ?    How  is  the  payment  generally 
made  ?     What  is  a  bill  of  exchange  ?     Who  is  the  drawer  ?     Who  the 
drawee  ?    Who  the  buyer  or  remitter  ? 


266  FOREIGN   BILLS. 

3.  The  person  to  whom  the  money  is  ordered  to  be  paid  is 
called  the  payee  ;  and 

4.  Any  person  who  purchases  a  bill  of  exchange  is  called 
the  buyer  or  remitter. 

274.  A  bill  of  exchange  is  called  an  inland  bill,  when  the 
drawer  and  drawee  both  reside  in  the  same  country  ;  and  when 
they  reside  in  different  countries,  it  is  called  a  foreign  bill. 

Exchange  is  said  to  be  at  par,  when  an  amount  at  the 
place  from  which  it  is  remitted  will  pay  an  equal  amount  at 
the  place  to  which  it  is  remitted.  Exchange  is  said  to  be  at 
a  premium,  or  above  par,  when  the  sum  to  be  remitted  will 
pay  less  at  the  place  to  which  it  is  remitted  ;  and  at  a  dis- 
count, or  below  par,  when  it  will  pay  more. 

EXAMPLES. 

1.  A  merchant  at  Chicago  wishes  to  pay  a  bill  in  New 
York  amounting  to  $3675,  and  finds  that  exchange  is  1J  per 
cent  premium  :  what  must  he  pay  for  his  bill? 

2.  A  merchant  in  Philadelphia  wishes  to  remit  to  Charles- 
ton $8756.50,  and  finds  exchange  to  be  1  per  cent  below  par  ; 
what  must  he  pay  for  the  bill  ? 

3.  A  merchant  in  Mobile  wishes  to  pay  in  New  York 
$6584,  and  exchange  is  2|  per  cent  premium  :  how  much 
must  he  pay  for  such  a  bill '{ 

4.  A  merchant  in  Boston  wishes  to  pay  in  New  Orleans 
$4653.75  ;  exchange  between  Boston  and  New  Orleans  is  1J 
per  cent  below  par  :  what  must  he  pay  for  a  bill  ? 

5.  A  merchant  in  New  York  has  $3690  which  he  wishes 
to  remit  to  Cincinnati ;  the  exchange  is  1  \  per  cent  below 
par  :  what  will  be  the  amount  of  his  bill  ? 

FOREIGN  BILLS. 

275.  A  Foreign  Bill  of  Exchange  is  one  in  which  the 
drawer  and  drawee  live  in  different  countries. 

NOTE. — In  all  Bills  of  Exchange  on  England,  the  £  sterling  is 
the  unit  or  base,  and  is  still  reckoned  at  its  former  value  of  $4$ 
=  $4.4444  -f,  instead  of  its  present  value  $4.84. 

274.  When  is  a  bill  of  exchange  said  to  be  inland  ?  When  foreign  ? 
When  is  exchange  said  to  be  at  par  ?  When  at  a  premium  ?  When 
at  a  discount  ? 


FOREIGN   BILLS.  267 

Hence,  £1  =$4.4444  -f 

Add  9  per  cent,  .3999 

Gives  the  present  value  of  £1  $4.8443. 

•Hence,  the  true  par  value  of  Exchange  on  England  is 
9  per  cent  on  the  nominal  base. 

1.  A  merchant  in  New  York  wishes  to  remit  to  England 
a'  bill  of  Exchange  for  £125  15s.  Qd  :  how  much  must  he 
pay  for  this  bill  when  exchange  is  at  9J  per  cent  premium? 

£125  15s.  6d.  ......  =£125.775 

Add  9|  per  cent  ..... 

gives  amount  in  £'s,  at  $4f== 


NOTE.  —  The  pounds  and  decimals  of  a  pound  are  reduced  to 
dollars  by  multiplying  by  40  and  dividing  by  9  —  giving,  in  this 
case,  $612.105. 

RULE.—  I.  Reduce  the  amount  of  the  bill  to  pounds  and 
decimals  of  a  pound,  and  then  add  the  premium  of  exchange. 

II.  Multiply  the  result  by  40  and  divide  the  product  by 
9  :  the  quotient  will  be  the  answer  in  United  States  Money. 

2.  A  merchant  shipped  100  bales  of  cotton  to  Liverpool, 
each  weighing  450  pounds.      They  were   sold  at  *l\d.  per 
pound,  and  the  freight  and  charges  amounted  to  £187   10s. 
He  sold  his  bill  of  exchange  at  9}  per  cent  premium  :  how 
much  should  he  receive  in  United  States  Money  ? 

3.  There  were  shipped  from  Norfolk,  Ya.,  to  Liverpool, 
Sbhhd.  of  tobacco,  each  weighing  450  pounds.     It  was  sold 
at  Liverpool  for  l^^d.  per  pound,  and  the  expenses  of  freight 
and  commissions  were  £92   Is.   Sd.     If  exchange  in  New 
York  is  at  a  premium  of  9J  per  cent,  what  should  the  owner 
receive  for  the  bill  of  exchange,  in  United  States  Money  ? 

276.  The  unit  or  base  of  the  French  Currency  is  the  French 
franc,  of  the  value  of  18  cents  6  mills.  The  franc  is  divided  into 
tenths,  called  decimes,  corresponding  to  our  dimes,  and  into 
centimes  corresponding  to  cents.  Thus,  5.12  is  read,  5  francs 
and  12  centimes. 

275.  What  is  a  foreign  bill  of  exchange  ?  In  bills  on  England,  wh.it 
is  the  unit,  or  base?  What  is  the  exchange  value  of  the  £  sterling  ? 
How  much  is  the  true  value  above  the  commercial  value  of  the  £  ster- 
ling? How  do  you  find  the  value  of  a  bill  in  English  currency  in 
United  States  mo'ney? 


268  DUTIES. 

All  bills  of  exchange  on  France  are  drawn  in  francs. 
Exchange  is  quoted  in  New  York  at  so  many  francs  and 
centimes  to  the  dollar. 

1.  What  will  be  the  value  of  a  bill  of  exchange  for  4^36 
francs,  at  5,25  to  the  dollar  ? 

ANALYSIS. — Since  1  dollar  will  buy 

5.25  francs,  the  bill  will  cost  as  many  OPERATION. 

dollars  as  5.25  is  contained  timesin  the     5.25)4536($864     Ans 
amount  of  the  bill ;  hence, 

Divide  the  amount  of  the  bill  by  the  value  of$l  in  francs: 
the  quotient  is  the  amount  to  be  paid  in  dollars. 

2.  What  will  be  the  amount  to  be  paid,  United  States 
money,  for  a  bill  of  exchange  on  Paris,  of  6530  francs, — 
exchange  being  5.14  francs  per  dollar  ? 

3.  What  will  be  the  amount  to  be  paid  in  United  States 
money  for  a  bill  of  exchange  on  Paris  of  10262  francs,  ex- 
change being  5.09  francs  per  dollar  ? 

4.  What  will  be  the  value  in  United  States  money  of  a 
bill  for  87595  francs,  at  5.16  francs  per  dollar? 

DUTIES. 

277.  Persons  who  bring  goods  or  merchandise  into  the 
United  States,  from  foreign  countries,  are  required  to  land 
them  at  particular  places  or  Ports,  called  Ports  of  Entry,  and 
to  pay  a  certain  amount  of  their  value,  called  a  Duty.  This 
duty  is  imposed  by  the  General  Government,  and  must  be 
the  same  on  the  same  articles  of  merchandise,  in  every  part 
of  the  United  States. 

Besides  the  duties  on  merchandise,  vessels  employed  in 
commerce  are  required,  by  law,  to  pay  certain  sums  for  the 
privilege  of  entering  the  ports.  These  sums  are  large  or 
small,  in  proportion  to  the  size  or  tonnage  of  the  vessels. 
The  moneys  arising  from  duties  and  tonnage,  are  called 
revenues. 

276.  What  is  the  unit  or  base  of  the  French  currency  ?     What  is  its 
value?    How  is  it  divided  ?    In  what  currency  arc  French  bills  of  ex- 
change drawn  ? 

277.  What  is  a  port  of  entry?  What  is  a  duty?    By  whom  are  duties 
imposed  ?    What  charges  are  vessels  required  to  pay  ?     What  are  the 
moneys  arising  from  duties  and  tonnage  called  ? 


DUTIES.  269 

278.  The  revenues  of  the  country  are  under  the  general 
direction  of  the  Secretary  of  the  Treasury,  and  to  secure  their 
faithful   collection,   the   government   has   appointed  various 
officers  at  each  port  of  entry  or  place  where  goods  may  be 
landed. 

279.  The  office  established  by  the  government  at  any  port 
of  entry  is  called  a  Custom  House,  and  the  officers  attached 
to  it  are  called  Custom  House  Officers. 

280.  All  duties  levied  by  law  on  goods  imported  into  the 
United  States,  are  collected  at  the  various  custom  houses,  and 
are  of  two  kinds,  Specific  and  Ad  valorem. 

A  specific  duty  is  a  certain  sum  on  a  particular  kind  of 
goods  named  ;  as  so  much  per  square  yard  on  cotton  or  wool- 
len cloths,  so  much  per  ton  weight  on  iron,  or  so  much  per 
gallon  on  molasses. 

An  ad  valorem  duty  is  such  a  per  cent  on  the  actual  cost 
of  the  goods  in  the  country  from  which  they  are  imported. 
Thus,  an  ad  valorem  duty  of  15  per  cent  on  English  cloth,  is 
a  duty  of  15  per  cent  on  the  cost  of  cloths  imported  from  Eng- 
land. 

281.  The  laws  of  Congress  provide,  that  the  cargoes  of  all 
vessels  freighted  with  foreign  goods  or  merchandise  shall  be 
weighed  or  gauged  by  the  custom  house  officers  at  the  port  to 
which  they  are  consigned.     As  duties  are  only  to  be  paid  on 
the  articles,  and  not  on  the  boxes,  casks  and  bags  which  con- 
tain them,  certain  deductions  are  made  from  the  weights  and 
measures,  called  Allowances. 

Gross  Weight  is  the  whole  weight  of  the  goods,  together 
with  that  of  the  hogshead,  barrel,  box,  bag,  &c.,  which  con- 
tains them. 

L ; __^____ 

278.  Under  whose  direction  are  the  revenues  of  the  country  ? 

279.  What  is  a  custom  house  ?    What  are  the  officers  attached  to  it 
called  ? 

280.  Where  are  the  duties  collected  ?    How  many  kinds  are  there, 
and  what  are  they  called  ?    What  is  a  specific  duty  ?    An  ad  valorem 
duty  ? 

281.  What  do  the  laws  of  Congress  direct  in  relation  to  foreign 
goods?    Why  are  deductions   made  from   their  weight?    What   are 
these  deductions  called  ?     What  is  gross  weight  ?    What  is  draft  ? 
What  is  the  greatest  draft  allowed  ? '   What  is  tare  ?     What  arc  the 
different  kinds  of  tare  ?    What  allowances  are  made  on  liquors  ? 


270  DUTIES. 

Draft  is  an  allowance  from  the  gross  weight  on  account  of 
waste,  where  there  is  not  actual  tare. 

On     112/6.  it  is  1/6. 

From     112  to    224  <      2, 

224  to    336  '      3, 

336  to  1120  '      4, 

1120  to  2016  '      7, 

Above  2016  any  weight  '      9  ; 
consequently,  9/6.  is  the  greatest  draft  allowed. 

Tare  is  an  allowance  made  for  the  weight  of  the  boxes, 
barrels,  or  bags  containing  the  commodity,  and  is  of  three 
kinds  :  1st,  Legal  tare,  or  such  as  is  established  by  law  ;  2d, 
Customary  tare,  or  such  as  is  established  by  the  custom  among 
merchants  ;  and  3c?,  Actual  tare,  or  such  as  is  found  by  re- 
moving the  goods  and  actually  weighing  the  boxes  or  casks 
in  which  they  are  contained. 

On  liquors  in  casks,  customary  tare  is  sometimes  allowed 
on  the  supposition  that  the  cask  is  not  full,  or  what  is  called 
its  actual  wants;  and  then  an  allowance  of  5  per  cent  for 
leakage. 

A  tare  of  10  per  cent  is  allowed  on  porter,  ale  and  beer,  in 
bottles,  on  account  of  breakage,  and  5  per  cent  on  all  other 
liquors  in  bottles.  At  the  custom  house,  bottles  of  the  com- 
mon size  are  estimated  to  contain  2J  gallons  the  dozen. 

NOTE. — For  table8  of  Tare  and  Duty,  see  Ogden  on  the  Tariff 
of  1842. 

EXAMPLES. 

1.  What  will  be  the  duty  on  125  cartons  of  ribbons,  each 
containing  48  pieces,  and  each  piece  weighing  802.  net,  and 
paying  a  duty  of  $2.50  per  pound  ? 

2.  What  will  be  the  duty  on  225  bags  of  coffee,  each  weigh- 
ing: gross  160/6.,  invoiced  at  6  cents  per  pound  ;  2  per  cent 
being  the  legal  rate  of  tare,  and  20  per  cent  the  duty  ? 

3.  What  duty  must  be  paid  on  275  dozen  bottles  of  claret, 
estimated  to  contain  2J  gallons  per  dozen,  5  per  cent  beinar 
allowed  for  breakage,  and  the  duty  being  35  cents  per  gallon? 

4.  A  merchant/ imports  175  cases  of  indigo,   each  case 
weighing  196/fo?.  gross  ;  15  per  cent  is  the  customary  rate  of 
tare,  and  the  duty  5  cents  per  pound  :  what  duty  must  he 
pay  on  the  whole  ? 


ALLIGATION   MEDIAL.  271 


ALLIGATION    MEDIAL. 

282.  ALLIGATION  MEDIAL  is  the  process  of  finding  the 
price  of  a  mixture  when  the  quantity  of  each  simple  and  its 
price  are  known. 

1.  A  merchant  mixes  Sib.  of  tea,  worth  75  cents  a  pound, 
with  16/6.  worth  $1.02  a  pound  :  what  is  the  price  of  the 
mixture  per  pound  ? 

ANALYSIS. — The  quantity,  8lb.  of  OPERATION. 

tea,  at  75  cents  a  pound,  costs  $6  ;  8/6.  at  75cte.=$   6  00 

and  16».   at    $1.03    costs   $16.32  :  16/6   at  $1  Q2  =  $16.32 

hence,  the   mixture,  =  24lb,,  costs  \ . 

$22.32 ;  and  the  price  of  lib.  of  the  24                       24)22.32 

mixture  is  found  by  dividing  this  $0  93 
cost  by  24  :  hence,  to  find  the  price  of  the  mixture, 

I.  Find  the  cost  of  the  entire  mixture  : 

II.  Divide  the  entire  cost  of  the  mixture  by  the  sum  of 
the  simples,  and  the  quotient  will  be  the  price  of  the  mixture. 

EXAMPLES. 

1.  A  farmer  mixes  30  bushels  of  wheat  worth  5s.  per 
bushel,  with   72  bushels  of  rye  at  3s.  per  bushel,  and  with 
60  bushels  of  barley  worth  2s.  per  bushel :  what  should  be 
the  price  of  a  bushel  of  the  mixture  ? 

2.  A  wine  merchant  mixes  15  gallons  of  wine  at  $1  per 
gallon  with  25  gallons  of  brandy  worth  75  ceuts  per  gallon  : 
what  should  be  the  price  of  a  gallon  of  the  compound  ? 

3.  A  grocer  mixes  40  gallons  of  whisky  worth  31  cents 
per  gallon  with  3  gallons  of  water  which  costs  nothing  :  what 
should  be  the  price  of  a  gallon  of  the  mixture  ? 

4.  A  goldsmith  melts  together  2/6.  of  gold  of  22  carats 
fine,  602:.  of  20  carats  fine,  and  6oz.  of  16  carats  fine  :  what 
is  the  fineness  of  the  mixture  ? 

5.  On  a  certain  day  the  mercury  in  the  thermometer  was 
observed  to  average  the  following  heights  :  from  6  in  the 
morning  to  9,  64°  ;  from  9  to  12,  74°  ;  from  12  to  3,  84°  ; 
and  from  3  to  6,  70°  :  what  was  the  mean  temperature  of 
the  day  ? 

282.  What  is  Alligation  Medial  ?    What  is  the  rule  for  determining 
the  price  of  the  mixture  ? 


272 


ALLIGATION   ALTERNATE. 


ALLIGATION  ALTERNATE. 

283.  ALLIGATION  ALTERNATE  is  the  process  of  finding  what 
proportions  must  be  taken  of  each  of  several  simples,  whose 
prices  are  known,  to  form  a  compound  of  a  given  price.     It 
is  the  opposite  of  Alligation  Medial,  and  may  be  proved  by  it. 

284.  To  find  the  proportional  parfe. 

1.  A  farmer  would  mix  oats  at  3s.  a  bushel,  rye  at  6s.,  and 
wheat  at  9s.  a  bushel,  so  that  the  mixture  shall  be  worth  5 
shillings  a  bushel  :  what  proportion  must  be  taken  of  each 
sort? 


OPERATION, 


oats,     3 


5  -j  rye, 


wheat,  9 


A. 


B. 


c. 

D. 

E. 

2 

1 

3 

2 

2 

1 

1 

ANALYSIS. — On  every  bushel  put  into  the  mixture,  whose  price 
is  less  than  the  mean  price,  there  will  be  a  gain  ;  on  every  bushel 
whose  price  is  greater  than  the  mean  price,  there  will  be  a  loss  ; 
and  since  there  is  to  be  neither  gain  nor  loss  by  the  mixture,  the 
gains  and  losses  must  balance  each  other. 

A  bushel  of  oats,  when  put  into  the  mixture,  will  bring  5  shil- 
lings, giving  a  gain  of  2  shillings ;  and  to  gain  1  shilling,  we  must 
take  half  as  much,  or  \  a  bushel,  which  we  write  in  column  A. 

On  1  bushel  of  wheat  there  will  be  a  loss  of  4  shillings  ;  and 
to  make  a  loss  of  1  shilling,  we  must  take  £  of  a  bushel,  which 
we  also  write  in  column  A :  i  and  £  are  called  proportional 
numbers. 

Again  :  comparing  the  oats  and  rye,  there  is  a  gain  of  2  shil- 
lings on  every  bushel  of  oats,  and  a  loss  of  1  shilling  on  every 
bushel  of  rye :  to  gain  1  shilling  on  the  oats,  we  take  \  a  bushel, 
and  to  lose  1  shilling  on  the  rye,  we  take  1  bushel :  these  num- 
bers are  written  in  column  B.  Two  simples,  thus  compared,  are 
called  a  couplet :  in  one,  the  price  of  unity  is  less  tJian  the  mean 
price,  and  in  the  other  it  is  greater. 

If,  every  time  we  take  i  a  bushel  of  oats  we  take  ^  of  a  bushel 
of  wheat,  the  gain  and  loss  will  balance  ;  and  if  every  time  we 
take  ^  a  bushel  of  oats  we  take  1  bushel  of  rye,  the  gain  and  loss 


283.  What  is  Alligation  Alternate  ? 

J284.  How  do  you  lind  the  proportional  numbers/* 


ALLIGATION   ALTERNATE. 


273 


will  balance :  hence,  if  tTie  proportional  numbers  of  a  couplet  be 
multiplied  by  any  number,  the  gain  and  loss  denoted  by  the  products, 
will  balance. 

When  the  proportional  numbers,  in  any  column,  are  fractional 
(as  in  columns  A  and  B),  multiply  them  by  the  least  common 
multiple  of  their  denominators,  and  write  the  products  in  new 
columns  C  and  D.  Then,  add  the  numbers  in  columns  C  and  D, 
standing  opposite  each  simple,  and  if  their  sums  have  a  common 
factor,  reject  it :  the  last  result  Will  be  the  proportional  numbers. 

RULE. — I.  Write  the  prices  or  qualities  of  the  simples  in  a 
column,  beginning  with  the  lowest,  and  the  mean  price  or 
quality  at  the  left. 

II.  Opposite  the  first  simple  write  the  part  which  must  be 
taken  to  gain  1  of  the  mean  price,  and  opposite  the  other  simple 
of  the  couplet,  write  the  part  which  must  be  taken  to  lose  1  of 
the  mean  price,  and  do  the  same  for  each  simple. 

III,  W  hen  the  proportional  numbers  are  fractional,  reduce 
them  to  integral  numbers,  and  then  add  those  which  stand  oppo- 
site the  same  single:  if  the  sums  have  a  common  factor,  reject 
it :  the  result  will  denote  the  proportional  parts. 

2.  A  merchant  would  mix  wines  worth  16s.,  18s.,  and  22s. 
per  gallon,  in  such  a  way,  that  the  mixture  may  be  worth 
20s.  per  gallon  :  what  are  the  proportional  parts  ? 


OPERATION.  . 

A. 

B. 

C. 

D. 

E. 

(161 

204l8J 
(22 

1 
1 

1 

i 

1 

1 
1 

1 
1 
3 

PROOF. 

1  gallon,  at  16  shillings,         ==  16s. 
1  gallon,  at  18  shillings,          =  18s. 
3  gallon,  at  22  shillings,          =  66s. 

5)  100 (2 Os.,  mean  price. 

N'OTE. — The  answers  to  the  last,  and  to  all  similar  questions, 
will  be  infinite  in  number,  for  two  reasons: 

1st.  If  the  proportional  numbers  in  column  E  be  multiplied  by 
any  number,  integral  or  fractional,  the  products  will  denote  pro- 
portional parts  of  the  simples. 

2d.  If  the  proportional  numbers  of  any  couplet  be  multiplied  by 
18 


274:  ALLIGATION   ALTERNATE. 

any  number,  the  gain  and  loss  in  that  couplet  will  still  balance, 
and  the  proportional  numbers  in  the  final  result  will  be  changed. 

3.  What  proportions  of  tea,  at  24  cents,  30  cents,  33  cents 
and  36  cents  a  pound,  must  be  mixed  together  so  that  the 
mixture  shall  be  worth  32  cents  a  pound  ? 

4.  What  proportions  of  coffee  at  IQcts.,  20cts.  and  28cfe. 
per  pound,  must  be  mixed  together  so  that  the  compound 
shall  be  worth  24ds.  per  pound  ? 

5.  A  goldsmith  has  gold  of  16,  of  18,  of  23,  and  of  24  carats 
fine  :  what  part  must  be  taken  of  each  so  that  the  mixture 
shall  be  21  carats  fine? 

6.  What  portion  of  brandy,  at  14s.  per  gallon,  of  old  Ma- 
deira, at  24s  per  gallon,  of  new  Madeira,  at  21s.  per  gallon, 
and  of  brandy,  at  10s.  per  gallon,  must  be  mixed  together  so 
that  the  mixture  shall  be  worth  18s.  per  gallon  ? 

285.    When  the  quantity  of  one  simple  is  given  : 

I.  How  much  wheat,  at  9s.  a  bushel,  must  be  mixed  with 
20  bushels  of  oats  worth  3  shillings  a  bushel,  that  the  mix- 
ture may  be  worth  5  shillings  a  bushel  ? 

ANALYSIS. — Find  the  proportional  numbers  :  they  are  2  and  1 ; 
hence,  the  ratio  of  the  oats  to  the  wheat  is  \  :  therefore,  there, 
must  be  10  bushels  of  wheat. 

RULE. — I.  Find  the  proportional  numbers,  and  write  the 
given  single  opposite  its  proportional  number. 

II.  Multiply  the  given  simple  by  the  ratio  which  its  propor- 
tional number  bears  to  each  of  the  others,  and  the  products 
will  denote  the  quantities  to  be  taken  of  each. 

EXAMPLES. 

1.  How  much  wine,  at  5s.,  at  5s.  Gd.,  and  6s.  per  gallon 
must  be  mixed  with  4  gallons,  at  4s.  per  gallon,  so  that  the 
mixture  shall  be  worth  5s.  4d.  per  gallon  ? 

2.  A  fanner  would  mix  14  bushels  of  wheat,  at  $1,20  per 
bushel,  with  rye  at  72c/s.,  barley  at  48c£s.,  and  oats  at  36c/s. : 
how  much  must  be  taken  of  each  sort  to  make  the  mixture 
worth  64  cents  per  bushel  ? 

3.  There  is  a  mixture  made  of  wheat  at  4s.  per  bushel, 
rye  at  3s.,  barley  at  2s.,  with  12  bushels  of  oats  at  l&d.  per 
bushel  :  how  much  is  taken  of  each  sort  when  the  mixture  is 
worth  3s.  Qd.  ? 


ALLIGATION  ALTERNATE.  275 

4.  A  distiller  would  mix  40^ro/.  of  French  brandy  at  12s. 
per  gallon,  with  English  at  Is.  and  spirits  at  4s.  per  gallon  : 
what  quantity  must  be  taken  of  each  sort  that  the  mixture 
may  be  afforded  at  8s.  per  gallon  ? 

286.   When  the  quantity  of  the  mixture  is  given. 

1.  A  merchant  would  make  up  a  cask  of  wine  containing 
50  gallons,  with  wine  worth  16s.,  18s.  and  22s.  a  gallon,  in 
such  a  way  that  the  mixture  may  be  worth  20s.  a  gallon  : 
much  must  he  take  of  each  sort  ? 


ANALYSIS.  —  This  is  the  same  as  example  2,  except  that  the 
quantity  of  the  mixture  is  given.  If  the  quantity  of  the  mixture 
be  divided  by  5,  the  sum  of  the  proportional  parts,  the  quotient 
10  will  show  how  many  times  each  pwportional  part  must  be  taken 
to  make  up  50  gallons  :  hence,  there  are  10  gallons  of  the  first, 
10  of  the  second,  and  30  of  the  third  :  hence, 

RULE.  —  I.  Find  the  proportional  parts. 

II.  Divide  the  quantity  of  the  mixture  by  the  sum  of  the 
proportional  parts,  and  the  quotient  will  denote  how  many 
times  each  part  is  to  be  taken.  Multiply  this  quotient  by 
the  parts  separately,  and  each  product  will  denote  the  quan- 
tity of  the  corresponding  simple. 

EXAMPLES. 

1.  A  grocer  has  four  sorts  of  sugar,  worth  12c?.,  Wd.,  6d 
and  4:d.  per  pound  ;  he  would  make  a  mixture  of  144  pounds 
worth  Sd.  per  pound  :  what  quantity  must  be  taken  of  each 
sort? 

2.  A  grocer  having  four  sorts  of  tea,  worth  5s.,  6s.,  8s.  and 
9s.  per  pound,  wishes  a  mixture  of  87  pounds  worth  7s,  per 
pound  :  how  much  must  he  take  of  each  sort  ? 

3.  A  silversmith  has  four  sorts  of  gold,  viz.,  of  24  carats 
fine,  of  22  carats  fine,  and  of  20  carats  fine,  and  of  15  carats  fine  : 
he  would  make  a  mixture  of  42oz.  of  17  carats  fine  ;  how 
much  must  be  taken  of  each  sort  ? 

PROOF.  —  All  the  examples  of  Alligation  Medial  may  be 
found  by  Alligation  Alternate. 

285.  How  do  you  find  the  quantity  of  each  simple  when  the  quantity 
of  one  simple  is  known  ? 

386.  How  do  you  find  the  quantity  of  each  simple  when  the  quantity 
of  each  mixture  is  known  ? 


276  INVOLUTION. 

INVOLUTION. 

287.  A  POWER  is  the  product  of  equal  factors.     The  equal 
factor  is  called  the  root  of  the  power. 

The  first  power  is  the  equal  factor  itself,  or  the  root : 
The  second  power  is  the  product  of  the  root  by  itself  : 
The  third  power  is  the  product  when  the  root  is  taken  3 
times  as  a  factor  : 

The  fourth  power,  when  it  is  taken  4  times  : 
The  fifth  power,  when  it  is  taken  5  times,  &c. 

288.  The  number  denoting  how  many  times  the  root  is 
taken  as  a  factor,  is  called  the  exponent  of  the  power.     It  is 
written  a  little  at  the  right  and  over  the  root :  thus,  if  the 
equal  factor  or  root  is  4, 

4=       4  the  1st  power  of  4. 

42— 4x4=     16  the  2d    power  of  4. 

43 _4X4X4—     64  the  3d    power  of  4. 

44  =4:  x  4  x  4  x  4  =  256  the  4th  power  of  4. 

45 .-4x4x4x4x4  — 1024  the  5th  power  of  4. 

INVOLUTION  is  the  process  of  finding  the  powers  of 'number 's. 

NOTES. — 1.  There  are  three  things  connected  with  every  power : 
1st,  The  root ;  2d,  The  exponent ;  and  3d,  The  power  or  result  of 
the  multiplication. 

2  In  finding  a  power,  the  root  is  always  the  1st  power;  hence, 
the  number  of  multiplications  is  1  less  than  the  exponent; 

RULE. — Multiply  the  number  by  itself  as  many  times  less 
1  as  there  are  units  in  the  exponent,  and  the  last  product 
will  be  the  power. 

EXAMPLES. 

Find  the  powers  of  tne  following  numbers  : 


1.  Square  of  1. 

2.  Square  of  J. 

3.  Cube  of  |. 

4.  Square  of  f . 

5.  Square  of  9. 

6.  Cube  of  12 

1.  3d  power  of  125. 

8.  3d  power  of  16 

9.  4th  power  of  9. 


10.  5th  power  of  16. 

11.  6th  power  of  20. 

12.  2d   power  of  225 

13.  Square  of  2167. 

14.  Cube  of  321 

15.  4th  power  of  215. 

16.  5th  power  of  906. 

17.  6th  power  of  9. 

18.  Square  of  36049. 


EVOLUTION.  277 

EVOLUTION. 

289,  EVOLUTION  is  the  process  of  finding  the  factor  when 
we  know  the  power. 

The  square  root  of  a  number  is  the  factor  which  multiplied 
by  itself  once  will  produce  the  number. 

The  cube  root  of  a  number  is  the  factor  which  multiplied 
by  itself  twice  will  produce  the  number. 

Thus,  6  is  the  square  root  of  36,  because  6  x  6=36  ;  and 
3  is  the  cube  root  of  27,  because  3  x  3  x  3=27. 

The  sign  V  is  called  the  radical  sign.  When  placed  be- 
fore a  number  it  denotes  that  its  square  root  is  to  be  ex- 
tracted. Thus,  1/36  =  6. 

We  denote  the  cube  root  by  the  same  sign  by  writing  3 
over  it :  thus,  v^  denotes  the  cube  root  of  27,  which  is 
equal  to  3.  The  small  figure  3,  placed  over  the  radical,  is 
called  the  index  of  the  root. 

'EXTRACTION  OF  THE  SQUARE  ROOT. 

290.  The  square  root  of  a  number  is  a  factor  which  mul- 
tiplied by  itself  once  will  produce  the  number.  To  extract 
the  square  root  is  to  find  this  factor*  The  first  ten  numbers 
and  their  squares  are 

1,       2,       3,       4,        5,        6,        Y,        8,        9,         10. 
1,       4,       9,      16,     25,      36,      49,      64,      81.       100. 
The  numbers  in  the  first  line  are  the  square  roots  of  those 
in    the    second.      The    numbers    1,  4,  9,  16,  25,   36,    &c. 
having  exact  factors,  are  called  perfect  squares. 

A  perfect  square  is  a  number  which  has  two  exact  factors 

NOTE. — The  square  root  of  a  number  less  than  100  will  be  less 
than  10,  while  the  square  root  of  a  number  greater  than  100  will 
be  greater  than  10. 

287.  What  is  a  power  ?    What  is  the  root  of  a  power?    What  is  the 
first  power  ?    What  is  the  second  power  ?    The  third  power  ? 

288.  What  is  the  exponent  of  the  power  ?    How  is  it  written  ?    What 
is  Involution  ?    How  many  things  are  connected  with  every  power  ? 
How  do  you  find  the  power  of  a  number  ? 

289.  What  is  Evolution?    What  is  the  square  root  of  a  number? 
What  is  the  cube  root  of  a  number  ?    How  do  you  denote  the  square 
root  of  a  number  ?    How  the  cube  root  ? 


278 


EXTRACTION   OF  THE    SQUARE   ROOT. 


291.  What  is  the  square  of  36=3  tens +  6  units? 


ANALYSIS. — 36=3  tens+6  units,  is  first 
to  be  taken  6  units'  time,  giving  62+3  x  6 : 
then  taking  it  3  tens'  times,  we  have 
3  x  6+32,  and  the  sum  is  32+2(3  x  6)+62 : 
that  is, 


3  +  6 
3  +  6 

3x6  +  6* 
32+3x6 


32+2(3x6)+6 

The  square  of  a  number  is  equal  to  the  square  of  the  tens, 
plus  twice  the  product  of  the  tens  by  the  units,  plus  the  square 
of  the  units. 

The  same  may  be  shown  by  the  figure : 

Let    the    line   AB  re-      F  30 

present  the  3  tens  or  30, 
and  BC  the  six  units. 

Let  AD  be  a  square 
on  AC,  and  AE  a  square 
on  the  ten's  line  AB. 

Then  ED  will  be  a 
square  on  the  unit  line 
6,  and  the  rectangle  EF 
will  be  the  product  of 
HE,  which  is  equal  to 
the  ten's  line,  by  IE, 
which  is  equal  to  the 
unit  line  Also,  the 
rectangle  BK  will  be  the 
product  of  EB,  which  is 
equal  to  the  ten's  line,  by 
the  unit  line  B  C.  But  the  whole  square  on  AC  is  made  up  of 
the  square  AE,  the  two  rectangles  FE  and  EC,  and  the  square 
ED. 

1.  Let  it  now  be  required  to  extract  the  square  root  of 
1296. 

ANALYSIS. — Since  the  number  contains  more  than  two  places  of 
figures,  its  root  will  contain  tens  and  units.  But  as  the  square  of 
one  ten  is  one  hundred,  it  follows  that  the  square  of  the  tens  of 
the  required  root  must  be  found  in  the  two  figures  on  the  left  of 
96.  Hence,  we  point  off  the  number  into  periods  of  two  figures 
each. 


30 
6 

180 

6 
6 
36 

30       E 
900  +  180  +  180  +  36=1296. 

30 
30 

30 
6 

900 

180 

30 


C 


290.  What  is  the  square  root   of   a,  number  ?      What  are  perfect 
squares  ?    How  many  are  there  between  1  and  100  ? 

291.  Into  what  parts  may  a  number  be  decomposed?    When  so  de- 
composed, what  is  its  square  equal  to  ? 


EXTRACTION  OF  THE  SQUARE  ROOT.     279 

We  next  find  the  greatest  square  contained  in        OPERATION. 
12,  which  is  3  tens  or  30.      We  then  square  3  1296(36 

tens  which  gives  9  hundred,  and  then  place  9  un-  ~ 

der  the  hundreds'  place,  and  subtract ,  this  takes 
away  the  square   of  the  tens,  and  leaves  396,  66)396 

which  is  twice  the  product  of  the  tens  by  the  units  395 

plus  the  square  of  the  units. 

If  now,  we  double  the  divisor  and  then  divide  this  remainder, 
exclusive  of  the  right  hand  figure,  (since  that  figure  cannot  enter 
into  the  product  of  the  tens  by  the  units)  by  it,  the  quotient  will 
be  the  units  figure  of  the  root.  If  we  annex  this  figure  to  the 
augmented  divisor,  and  then  multiply  the  whole  divisor  thus  in- 
creased by  it,  the  product  will  be  twice  the  tens  by  the  units  plus 
the  square  of  the  units  ;  and  hence,  we  have  found  both  figures  of 
the  root. 

This  process  may  also  be  illustrated  by  the  figure. 

Subtracting  the  square  of  the  tens  is  taking  away  the  square 
AE  and  leaves  the  two  rectangles  FE  and  BK,  together  with  the 
Bquare  ED  on  the  unit  line. 

The  two  rectangles  FE  and  BK*representing  the  product  of  units 
by  tens,  can  be  expressed  by  no  figures  less  than  tens. 

If,  then,  we  divide  the  figures  39,  at  the  left  of  6,  by  twice  the 
tens,  that  is,  by  twice  AB  or  BE,  the  quotient  will  be  BG  or  EK 
the  unit  of  the  root. 

Then,  placing  BC  or  G,  in  the  root,  and  also  annexing  it  to  the 
divisor  doubled,  and  then  multiplying  the  whole  divisor  66  by  6, 
we  obtain  the  two  rectangles  FE  and  CE,  together  with  the 
equare  ED. 

292.  Hence,  for  the  extraction  of  the  square  root,  we  have 
the  following 

RULE. — I.  Separate  the  given  number  into  periods  of  two 
figures  each,  by  setting  a  dot  over  the  place  of  units,  a  se- 
cond over  the  place  of  hundreds,  and  so  on  for  each  alternate 
figure  at  the  left. 

II.  Note  the  greatest  square  contained  in  the  period  on 
the  left,  and  place  its  root  on  the  right  after  the  manner  of 
a  quotient  in  division.  Subtract  the  square  of  this  root 
from  the  first  period,  and  to  the  remainder  bring  down  the 
second  period  for  a  dividend. 

292.  What  is  the  first  step  in  extracting  the  square  root  of  numbers  ? 
What  is  the  second?  What  is  the  third?  What  the  fourth?  What 
the  fifth  ?  Give  the  entire  rule. 


280  EXTRACTION   OF   THE   SQUARE    ROOT. 

III.  Double  the  root  thus  found  for  a  trial  divisor  and 
place  it  on  the  left  of  the  dividend.  Find  how  many 
times  the  trial  divisor  is  contained  in  the  dividend,  exclu- 
sive of  the  right-hand  figure,  and  place  the  quotient  in  the 
root  and  also  annex  it  to  the  divisor. 

IY.  Multiply  the  divisor  thus  increased,  by  the  last  figure 
of  the  root ;  subtract  the  product  from  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  new  divi- 
dend. 

Y.  Double  the  ivhole  root  thus  found,  for  a  new  trial  di- 
visor, and  continue  the  operation  as  before,  until  all  the 
periods  are  brought  down. 

EXAMPLES. 

1.  What  is  the  square  root  of  263169  ? 

OPERATION. 

ANALYSIS. — We  first  place  a  dot  over  the  «  a   o'i    A  6  /  K  i  Q 

9,  making  the  right-hand  period  69.  We 
then  put  a  dot  over  the  1  and  also  over  the 
6,  making  three  periods.  101)131 

The  greatest  perfect  square  in  26  is  25,  AI 

the  root  of  which  is  5,     Placing  5  in  the 


root,  subtracting  its  square  from  26,  and        1023)3069 
bringing  down  the  next  period  31,  we  have  3069 

131  for  a  dividend,  and  by  doubling  the 

root  we  have  10  for  a  trial  divisor.  Now,  10  is  contained  in  13, 
1  time.  Place  1  both  in  the  root  and  in  the  divisor :  then  multi- 
ply 101  by  1 ;  subtract  the  product  and  bring  down  the  next  period. 
We  must  now  double  the  whole  root  51  for  a  new  trial  divisor  ; 
or  we  may  take  the  first  divisor  after  having  doubled  the  last 
figure  1 ;  then  dividing,  we  obtain  3,  the  third  figure  of  the  root. 

NOTE. — 1.  The  left-hand  period  may  contain   but  one  figure; 
each  of  the  others  will  contain  two. 

2.  If  any  trial  divisor  is  greater  than  its  dividend,  the  corres 
ponding  quotient  figure  will  be  a  cipher. 

3.  If  the  product  of  the  divisor  by  any  figure  of  the  root  exceeds 
the  corresponding  dividend,  the  quotient  figure  is  too  large  and 
must  be  diminished. 

4.  There  will  be  as  many  figures  in  the  root  as  there  are  periods 
in  the  given  number. 

5.  If  the  given  number  is  not  a  perfect  square  there  will  be  a 
remainder  after  all  the  periods  are  brought  down.     In  this  case, 
periods  of  ciphers  may  be  annexed,  forming  new  periods,  each  of 
which  will  give  one  decimal  place  in  the  root. 


EXTRACTION   OF   THE   SQUARE    ROOT. 


281 


What  is  the  square  root  of  36729  :        OPERATION. 

3  67  29(191.64  + 
1 


In  this  example  there  are  two 
periods  of  decimals,  which  give  two 
places  of  decimals  in  the  root. 


29)267 
261 

381)629 
381 


3826)24800 
22956 


38324)184400 
153296 
31104  Hem. 


293.  To  extract  the  square  root  of  a  fraction. 


1.  What  is  the  square  root  of  .5  ? 


NOTE. — We  first  annex  one  cipher  to 
make  even  decimal  places.  We  then  ex- 
tract the  root  of  the  first  period  :  to  the 
remainder  we  annex  two  ciphers,  forming 
a  new  period,  and  so  on. 


OPERATION. 

.50(.707  + 
49 

140)100 
000 


1407)10000 
9849 


151  Rem. 


OPERATION. 


2.  What  is  the  square  root  of  £  ? 

NOTE.  —  The  square  root  of  a  fraction 
is  equal  to  the  square  root  of  the  numerator 
divided  by  the  square  root  of  the  denomi- 
nator. 


3.  What  is  the  square  root  of  J  ? 

NOTE. — When  the  terms  are  not  per- 
fect squares,  reduce  the  common  fraction  |  = .  7  5  ; 
to  a  decimal  fraction,  and  then  extract  x/sZrv/VcT_ 
the  square  root  of  the  decimal.  5          •  *&  —  • 


OPERATION. 


293.  How  do  you  extract  the  square  root  of  a  decimal  fraction  ? 
ef  a  common  fraction  ? 


How 


282 


SQUARE   ROOT. 


RULE. — I.  If  ike  fraction  is  a  decimal,  point  off  the 
periods  from  the  decimal  point  to  the  right,  annexing  ci- 
phers if  necessary,  so  that  each  period  shall  contain  two 
places,  and  then  extractJhe  root  as  in  integral  numbers. 

II.  If  the  fraction  is  a  common  fraction,  and  its  terms 
perfect  squares,  extract  the  square  root  of  the  numerator  and 
denominator  separately  ;  if  they  are  not  perfect  squares,  re- 
duce the  fraction  to  a  decimal,  and  then  extract  the  square 
root  of  the  result. 

EXAMPLES. 

What  are  the  square  roots  of  the  following  numbers  ? 


of  3? 
of  11? 
of  1069  ? 
of  2268741? 


5.  of  7596796? 


of  36372961? 
of  22071204? 
of  3271.4207? 
of  4795.25731? 


10.  of  4.372594? 


11.  of  .0025? 

12.  of  .00032754? 

13.  of  .00103041? 

14.  of  4.426816? 

15.  of8f  ? 

16.  of  9J? 

17.  of^? 

18.  o 

19.  o 

20.  off 


APPLICATIONS    IN    SQUARE    ROOT. 


294.  A  triangle  is  a  plain  figure  which  has  three  sides  and 
three  angles. 


If  a  straight  line  meets  another  straight  line, 
making  the  adjacent  angles  equal,  each  is 
called  a  right  angle  ;  and  the  lines  are  said 
to  be  perpendicular  to  each  other. 

295.  A  right  angled  triangle  is  one 
which  has  one  right  angle.  In  the  right 
angled  triangle  ABC,  the  side  AC  opposite 
the  right  angle  B  is  called  the  hi/pothenuse ; 
the  side  AB  the  base;  and  the  side  BC 
the  perpendicular. 


APPLICATIONS. 


283 


29G.  In  a  right  angled  triangle  the  square  described  in 
the  hypothemise  is  equal  to  the  sum  of  the  squares  described 
in  the  other  two  sides. 

Thus,  if  AC13  be  a  right 
angled  triangle,  right  an- 
gled at  C,  then  will  the 
large  square,  D,  described 
on  the  hypothenuse  AB,  be1 
equal  to  the  sum  of  the 
squares  F  and  E  described 
on  the  sides  AC  and  CB. 
This  is  called  the  carpen- 
ter's theorem.  By  count- 
ing the  small  squares  in  the 
large  square  D,  you  will 
find  their  number  equal 
to  that  contained  in  the 

small  squares  F  and  E.  In  this  triangle  the  hypothenuse 
AB  =  5,  AC  =  4,  and  CB  =  3.  Any  numbers  having  the 
same  ratie,  as  5,  4  and  3,  such  as  10,  8  and  6  ;  20,  16  and 
12,  &c.,  will  represent  the  sides  of  a  right  angled  triangle. 


1.  Wishing  to  know  the  distance  from  A 
to  the  top  of  a  tower,  I  measured  the  height 
of  the  tower  and  found  it  to  be  40  feet  ;  also 
the  distance  from  A  to  B  and  found  it  30  feet  ; 
what  was  the  distance  from  A  to  C  ? 
302=  900 


BC=40;   BC^402^ 


~       2500 


=  ^2500  =  50  feet. 


297.  Hence,  when  the  base  and  perpendicular  are  known 
and  the  hypothenuse  is  required, 


294.  What  is  a  triangle  ?    What  is  a  right  angle  ? 

295.  What  is  a  right  angled  triangle  ?    Which  side  is  the  hypothe- 
nuse ? 

296.  In  a  right  angled  triangle  what  is  the  square  on  the  hypothe- 
nuse equal  to  ? 


284  SQUARE   ROOT. 

Square  the  base  and  square  the  perpendicular,  add  the  re- 
sults and  then  extract  the  square  root  of  their  sum. 

2.  What  is  the  length  of  a  rafter  that  will  reach  from  the 
eaves  to  the  ridge  pole  of  a  house,  when  the  height  of  the 
roof  is  15  feet  and  the  width  of  the  building  40  feet  ? 

298.  To  find  one  side  when  we  know  the  hypothenuse  and 
the  other  side. 

3.  The  length  of  a  ladder  which  will  reach  from  the  mid- 
dle of  a  street  80  feet  wide  to  the  eves  of  a  house,  is  50  feet : 
what  is  the  height  of  the  house  ?  Ans.  30  feet. 

ANALYSIS — Since  the  square  of  the  length  of  the  ladder  is  equal 
to  the  sum  of  the  squares  of  half  the  street  and  the  height  of  the 
house,  the  square  of  the  length  of  the  ladder  diminished  by  the 
square  of  half  the  street  will  be  equal  to  the  square  of  the  height 
of  the  house  :  hence, 

Square  the  hypothenuse  and  the  known  side,  and  take  the 
difference  ;  the  square  root  of  the  difference  will  be  the  other 
side. 

EXAMPLES. 

1.  If  an  acre  of  land  be  laid  out  in  a  square  form,  what 
will  be  the  length  of  each  side  in  rods  ? 

2.  What  will  be  the  length  of  the  side  of  a  square,  in  rods, 
that  shall  contain  100  acres  ? 

3.  A  general  has  an  army  of  7225  men  :  how  many  must 
be  put  in  each  line  in  order  to  place  them  in  a  square  form  ? 

4.  Two  persons  start  from  the  same  point ;  one  travels 
due  east  50  miles,  the  other  due  south  84  miles  :  how  far  are 
they  apart  ? 

5.  What  is  the  length,  in  rods,  of  one  side  of  a  square  that 
shall  contain  12  acres  ? 

6.  A  company  of  speculators  bought  a  tract  of  land  for 
$6724,  each  agreeing  to  pay  as  many  dollars  as  there  were 
partners  :  how  many  partners  were  there  ? 

297.  How  do  you  find  the  hypothenuse  when  you  know    the  base 
and  perpendicular  ? 

298.  If  you  know  the  hypothenuse  and  one  side,  how  do  you  find  the 
other  side  ? 


CUBE   ROOT.  285 

7.  A  farmer  wishes  to  set  out  an  orchard  of  3844  trees,  so 
that  the   number  of  rows  shall  be  equal  to  the  number  of 
trees  in  each  row  :  what  will  be  the  number  of  trees  ? 

8.  How  many  rods  of  fence  will  enclose  a  square  field  of 
10  acres  ? 

9.  If  a  line   150  feet  long  will  reach  from  the  top  of  a 
steeple  120  feet  high,  to  the  opposite  side  of  the  street,  what 
is  the  width  of  the  street  ? 

10.  What  is  the  length  of  a  brace  whose  ends  are  each  3| 
feet  from  the  angle  made  by  the  post  and  beam  ? 

CUBE    ROOT. 

299.  The  CUBE  ROOT  of  a  number  is  one  of  three  equal 
factors  of  the  number. 

To  extract  the  cube  root  of  a  number  is  to  find  a  factor 
which  multiplied  into  itself  twice,  will  produce  the  given 
number. 

Thus,  2  is  the  cube  root  of  8  ;  for,  2  x  2  x  2  =  8  :  and  3  is 
the  cube  toot  of  27  ;  for  3  x  3  x  3  =  27. 

1,       2,       3,       4,         5,         6,         7,         8,         9. 

1        8       27      64       125      216      343     512       729. 

The  numbers  in  the  first  line  are  the  cube  roots  of  the 
corresponding  numbers  of  the  second.  The  numbers  of  the 
second  line  are  called  perfect  cubes.  By  examining  the  num- 
bers of  the  two  lines  we  see, 

1st.  That  the  cube  of  units  cannot  give  a  higher  order  than 
hundreds. 

2d.  That  since  the  cube  of  one  ten  (10)  is  1000  and  the 
cube  of  9  tens  (90),  81000,  the  cube  of  tens  will  not  give  a 
lower  denomination  than  thousands,  nor  a  higher  denomi- 
nation than  hundreds  of  thousands. 

Hence,  if  a  number  contains  more  than  three  figures,  its 
cube  root  will  contain  more  than  one  :  if  it  contains  more 
than  six,  its  root  will  contain  more  than  two,  and  so  on  ; 
every  additional  three  figures  giving  one  additional  figure  in 
the  root,  and  the  figures  which  remain  at  the  left  hand, 
although  less  than  three,  will  also  give  a  figure  in  the  root, 
This  law  explains  the  reason  for  pointing  off  into  periods  of 
three  figures  each. 


286  CUBE  BOOT. 

300.  Let  us  now  see  how  the  cube  of  any  number,  as  16, 
is  formed.  Sixteen  is  composed  of  1  ten  and  6  units,  and 
may  be  written  10  -f  G.  To  nod  the  cube  of  16,  or  of  10+6, 
we  must  multiply  the  number  by  itself  twice 

To  do  this  we  place  the  number  thus  16=10-}-  6 

10+  6 

product  by  the  units  -  60+36 

product  by  the  tens  -100+  60 

Square  of  16  -  100+  120--*-  36 

Multiply  again  by  16  -  -  10+6 

product  by  the  units  -  600+  720+216 

product  by  the  tens  1000+1200+  360 

Cube  of  1 6  TOOO+T800  + 1080  +  2l6 

1.  By  examining  the  parts  of  this  number  it  is  seen  that 
the  first  part  1000  is  the  cube  of  the  tens  ;  that  is, 

10x10x10=1000. 

2.  The  second  part  1800  is  three  times  the  square  of  the 
tens  multiplied  by  the  units  ;  that  is, 

3  x  (10)*  x  6=3  x  100  x  6=1800. 

3.  The  third  part  1080  is  three  times  the  square  of  the  units 
multiplied  by  the  tens  ;  that  is, 

3  x62x  10=3x36x10=1080. 

4.  The"  fourth  part  is  the  cube  of  the  units  ;  that  is, 

63=6x  6x6=210. 
1.  What  is  the  cube  root  of  the  number  4096  ? 

ANALYSTS.— Since  the  number 

contains  more  than  three  figures,  4   096(16 

•we  iuaow  that  the  root  will  con-  1 

tain  at  least  units  and  tens.  ia  o  \1T~n — 7c\  Q    T   R 

.     Separating    the    three   right-  l*X_3  =  o)3   I 
hand    figures    from   the  4,   we  163=4   096 

know  that  the  cube  of  the  tens 
\vili  be  found  in  the  4  ;  and  1  is  the  greatest  cube  in  4. 

299.  What  is  the  cube  root  of  a  number  ?  How  many  perfect  cubes 
arc  there  between  1  and  1000  ?  Tin,.* 

800.  Of  how  many  parts  is  the  cube  of  a  number  composed  ?  What 
are  they  ? 


CUBE   BOOT.  287 

Hence,  we  place  the  root  1  on  the  right,  and  this  is  the  tens  of 
the  required  root.  We  then  cube  1  and  subtract  the  result  from 
4,  and  to  the  remainder  we  bring  down  the  first  figure  0  of  the 
next  period. 

We  have  seen  that  the  second  part  of  the  cube  of  16,  viz.  1800, 
is  three  times  the  square  of  the  tens  multiplied  by  the  units  :  and 
hence,  it  can  have  no  significant  figure  of  a  less  denomination  than 
hundreds.  It  must,  therefore,  make  up  a  part  of  the  30  hundreds 
above.  But  this  30  hundreds  also  contains  all  the  hundreds 
which  come  from  the  3d  and  4th  parts  of  the  cube  of  16.  If  it 
were  not  so,  the  30  hundreds,  divided  by  three  times  the  square 
of  the  tens,  would  give  the  unit  figure  exactly 

Forming  a  divisor  of  three  times  the  square  of  the  tens,  we  find 
the  quotient  to  be  ten  ,  but  this  we  know  to  be  too  large.  Placing 
9  in  the  root  and  cubing  19,  we  find  the  result  to  be  6859.  Then 
trying  8  we  find  the  cube  of  18  still  too  large  ;  but  when  we  take 
6  we  find  the  exact  number.  Hence  the  cube  root  of  4096  is  16. 

301.  Hence,  to  find  the  cube  root  of  a  number, 

RULE. — I.  Separate  the  given  number  into  periods  of  three 
figures  each,  by  placing  a  dot  over  the  place  of  units,  a  second 
over  the  place  of  thousands,  and  so  on  over  each  third  figure 
to  the  left ;  the  left  hand  period  will  often  contain  less  than 
three  places  of  figures. 

IT.  Note  the  greatest  perfect  cube  in  the  first  period,  and 
set  its  root  on  the  right,  after  the  manner  of  a  quotient  in  di- 
vision. Subtract  the  cube  of  this  n  umber  from  the  first  period, 
and  to  the  remainder  bring  down  the  first  figure  of  the  next 
period  for  a  dividend. 

III.  Take  three  times  the  square  of  the  root  just  found  for 
a  trial  divisor,  and  see  how  often  it  is  contained  in  the  divi- 
dend, and  place  the  quotient  for  a  second  figure  of  the  root. 
Then  cube  the  figures  of  the  root  thus  found,  and  if  their 
cube  be  greater  than  the  first  two  periods  of  the  given  num- 
ber, diminish  the  last  figure,  but  if  it  be  less,  subtract  it 
from  the  first  two  periods,  and  to  the  remainder  bringdown 
the  first  figure  of  the  next  period  for  a  new  dividend. 

IY.  Take  three  times  the  square  of  the  whole  root  for  a 
second  trial  divisor,  and  find  a  third  figure  of  the  root. 
Cube  the  whole  root  thus  found  and  subtract  the  result  from 
the  first  three  periods  of  the  given  number  when  it  is  less 
than  that  number,  but  if  it  is  greater,  diminish  the  figure 
of  the  root  /  proceed  in  a  similar  way  for  all  the  periods. 


288  CUBE   ROOT. 

EXAMPLES. 

1.  What  is  the  cube  root  of  99252841  ? 

99  252  847(463 
43=64 

4?  x  3=48)352     dividend. 
First  two  periods  99  252 

(46)*=46x  46x46=  97  336 

3  x  (46)2=634S  )       19T68  2d  dividend. 
The  first  three  periods      -  99  252  847 

(463)3         =99  252  847 
Find  the  cube  roots  of  the  following  numbers  : 


1.  Of  389017? 

2.  Of  5735339? 

3.  Of  32461759? 


4.  Of  84604519? 

5.  Of  259694072? 

6.  Of  48228544? 


302.  To  extract  the  cube  root  of  a  decimal  fraction. 

Annex  ciphers  to  the  decimal,  if  necessary,  so  that  it 
shall  consist  of  3,  6,  9,  &c.,  places.  Then  put  the  first  point 
over  the  place  of  thousandths,  the  second  over  the  place  of 
millionths,  and  so  on  over  every  third  place  to  the  right ; 
after  which  extract  the  root  as  in  whole  numbers. 

NOTES. — 1.  There  will  be  as  many  decimal  places  in  the  root 
as  there  are  periods  in  the  given  number. 

2.  The  same  rule  applies  when  the  given  number  is  composed 
of  a  whole  number  and  a  decimal. 

3.  If  in  extracting  the  root  of  a  number  there  is  a  remainder 
after  all  the  periods  have  been  brought  down,  periods  of  ciphers 
may  be  annexed  by  considering  them  as  decimals. 

EXAMPLES. 

Find  the  cube  roots  of  the  following  numbers.: 


1.  Of  .157464? 
2.  Of  .870983875  ? 
3.  Of  12.977875? 

4.  Of  .751089429? 
f>.   Of  .353393243  ? 
6.  Of  3.408862625? 

301.  What  is  the  rule  for  extracting  the  cube  root  ? 

303.  How  do  you  extract  the  cube  root  of  a  decimal  fraction  ?  How 
many  decimal  places  will  there  be  in  the  root  ?  Will  the  same  rulft 
apply  when  there  is  a  whole  number  and  a  decimal  ?  If  in  extracting 
the  root  of  any  number  you  find  i  decimal,  how  do  you  proceed  ? 


APPLICATIONS.  289 

303.  To  extract  the  cube  root  of  a  common  fraction. 

I.  Reduce  compound  fractions  to  simple  ones,  mixed  num- 
bers to  improper  fractions,  and  then  reduce  the  fraction  to 
its  lowest  terms. 

II.  Extract  the  cube  root  of  the  numerator  and  denomi- 
nator separately,  if  they  have  exact  roots  ;  but  if  either  of 
them  has  not  an  exact  root,  reduce  the  fraction  to  a  decimal 
and  extract  the  root  as  in  the  last  case, 

EXAMPLES. 

Find  the  cube  roots  of  the  following  fractions  : 

1.  Offf|?  4.  Of£? 

2.  Of31J&?  5.  Off? 
3-  OfT3^?                              6.  Of  |? 

APPLICATIONS. 

1.  What  must  be  the  length,  depth,  and  breadth  of  a  box, 
when  these  dimensions  are  all  equal  and  the  box  contains 
4913  cubic  feet  ? 

2.  The  solidity  of  a  cubical  block  is  21952  cubic  yards  : 
what  is  the  length  of  each  side  ?     What  is  the  area  of  the 
surface  ? 

3.  A  cellar  is  25  feet  long  20  feet  wide,  and  8|  feet  deep : 
what  will  be  the  dimensions  of  another  cellar  of  equal  capacity 
in  the  form  of  a  cube  ? 

4.  What  will  be  the  length  of  one  side  of  a  cubical  granary 
that  shall  contain  2500  bushels  of  grain  ? 

5.  How  many  small  cubes  of  2  inches  on  a  side  can  be 
sawed  out  of  a  cube  2  feet  on  a  side,  if  nothing  is  lost  in 
sawing  ? 

6.  What  will  be  the  side  of  a  cube  that  shall  be  equal  to 
the  contents  of  a  stick  of  timber  containing  1728  cubic  feet? 

7.  A  stick  of  timber  is  54  feet  long  and  2  feet  square  : 
what  would  be  its  dimensions  if  it  had  the  form  of  a  cube  ? 

NOTES. — 1.  Bodies  are  said  to  be  similar  when  their  like  parts 
are  proportional. 

2.  It  is  found  that  the  contents  of  similar  bodies  are  to  each 
other  as  the  cubes  of  their  like  dimensions. 

303.  How  do  you  extract  the  cube  root  of  a  vulgar  fraction  ? 
19 


290  ARITHMETICAL   PROGRESSION, 

3,  All  bodies  named  in  the  examples  are  supposed  to  be  simi 
lar. 

8.  If  a  sphere  of  4  feet  in  diameter  contains  33.5104  cubic 
feet,  what  will  be  the  contents  of  a  sphere  8  feet  in  diameter  ? 

43    :    83    :    :    33.5104    :  Am. 

9.  If  the  contents  of  a  sphere  14  inches  in  diameter  is 
1436.7584  cubic  inches,  what  will  be  the  diameter  of  a  sphere 
which  contains  11494.0672  cubic  inches  ? 

10.  If  a  ball  weighing  32  pounds  is  6  inches  in  diameter, 
what  will  be  the  diameter  of  a  ball  weighing  2048  pounds  ? 

11.  If  a  haystack,  24  feet  in  height,  contains  8  tons  of  hay, 
what  will  be  the  height  of  a  similar  stack  that  shall  contain 
but  1  ton  ? 

ARITHMETICAL   PROGRESSION. 

304.  An  Arithmetical  Progression  is  a  series  of  numbers  in 
which  each  is  derived  from  the  preceding  one  by  the  addition 
or  subtraction  of  the  same  number. 

The  number  added  or  subtracted  is  called  the  common  dif- 
ference. 

305.  If  the  common  difference  is  added,  the  series  is  called 
an  increasing  series. 

Thus,  if  we  begin  with  2,  and  add  the  common  difference, 
3,  we  have 

2,  5,  8,  11,  14,  17,  20,  23,  &c., 

which  is  an  increasing  series. 

If  we  begin  with  23,  and  subtract  the  common  difference, 
3,  we  hare 

23,  20,   17,  14,  11,  8,  5,  &c., 
which  is  a  decreasing  series. 

304.  What  is  an  arithmetical  progression  ?    What  is  the  number 
added  or  subtracted  called? 

305.  When  the  common  difference  is  added,  what  is  the  scries  called  ? 
What  is  it  called  when  the  common  difference  is  subtracted  ?    What 
are  the  several  numebrs  called  ?    What  arc  the  first  and  last  called  ? 
What  arc  the  intermediate  ones  called  ? 


ARITHMETICAL   PROGRESSION.  291 

The  several  numbers  are  called  the  terms  of  the  progres- 
sion or  series  :  the  first  and  last  are  called  the  extremes,  and 
the  intermediate  terms  are  called  means. 

306.  In  every   arithmetical    progression   there   are   five 
parts  : 

1st,  the  first  term  ; 

2d,    the  last  term  ; 

3d,    the  common  difference  ; 

4th,  the  number  of  terms  ; 

5th,  the  sum  of  all  the  terms. 

If  any  three  of  these  parts  are  known  or  given,  the  remain- 
ing ones  can  be  determined. 

CASE   I. 

307.  Knowing  the  first  term,  the  common  difference,  and 
the  number  of  terms,  to  find  the  last  term. 

1.  The  first  term  is  3,  the  common  difference  2,  and  the 
number  of  terms  19  :  what  is  the  last  term  ? 

ANALYSIS. — By  considering  the  manner  in 
which  the  increasing  progression  is  formed,  we 
see  that  the  2d  term  is  obtained  by  adding  the 
common  difference  to  the  1st  term ;  the  3d,  by          OPEBATION. 
adding  the  common  difference  to  the  2d ;  the        1 8  No.       less  1 
4th,  by  adding  the  common  difference  to  the          cj  Com     dif 
3d,  and  so  on  ;  the  number  of  additions  being  1        — 
less  than  the  number  of  terms  found.  35 

But  instead  of  making  the  additions,  we  may          3   1st  term, 
multiply  the  common  difference  by  the  number        ^7:  ,     ,   , 
of  additions,  that  is,  by  1  less  than  the  number  m 

of  terms,  and  add  the  first  term  to  the  pro- 
duct:  hence, 

RULE. — Multiply  the  common  difference  by  1  less  than 
the  number  of  terms  ;  if  the  progression  is  increasing,  add 
the  product  to  the  first  term  and  the  sum  ivill  be  the  last 
term ;  if  it  is  decreasing,  subtract  the  product  from  the 
first  term  and  the  difference  will  be  the  la?t  term. 

306.  How  many  parts  are  there  in  every  arithmetical  progression  ? 
What  are  they  ?  How  many  parts  must  be  given  before  the  remaining 
ones  can  be  found  ? 


292  ARITHMETICAL  PROGRESSION. 


EXAMPLES. 

1.  A  man  bought  50  yards  of  cloth,  for  which  he  was  tQ 
pay  6  cents  for  the  1st  yard,  9  cents  for  the  2d,  12  cents  for 
the  3d,  and  so  on  increasing  by  the  common  difference  3  : 
how  much  did  he  pay  for  the  last  yard  ? 

2.  A  man  puts  out  $100  at  simple  interest,  at  1  per  cent : 
at  the  end  of  the  1st  year  it  will  have  increased  to  $107,  at 
the  end  of  the  2d  year  to  $114,  and  so  on,  increasing  $t 
each  year  :  what  will  be  the  amount  at  the  end  of  1 6  years  ? 

3.  What  is  the  40th  term  of  an  arithmetical  progression  of 
which  the  first  term  is  1,  and  the  common  difference  1  ? 

4.  What  is  the  30th  term  of  a  descending  progression  of 
which  the  first  term  is  60,  and  the  common  difference  2  ? 

5.  A  person  had  35  children  and  grandchildren,  and  it  so 
happened  that  the  difference  of  their  ages  was  18  months, 
and  the  age  of  the  eldest  was  60  years  :  how  old  was  the 
youngest  ? 

CASE    II. 

308.  Knowing  the  two  extremes  and  the  number  of  terms, 
to  find  the  common  difference. 

1.  The  extremes  of  an  arithmetical  progression  are  8  and 
104,  and  the  number  of  terms  25  :  what  is  the  common  dif- 
ference ? 

ANALYSIS. — Since  the  common  difference 
multiplied  by  1  less  than  the  number  of  OPERATION. 

terms  gives  a  product  equal  to  the  differ  104 

erence  of  the  extremes,  if  we  divide  the  dif  g 

ference  of  the  extremes  by  1  less  than  the 


number  of  terms,  the  quotient  will  be  the         25—  1  —24)96(4. 
common  difference :  hence, 

RULE. — Subtract  the  less  extreme  from  the  greater  and 
divide  the  remainder  by  1  less  than  the  number  of  terms; 
the  quotient  will  be  the  common  difference. 

307.  "When  you  know  the  first  term,  the  common  difference,  and  the 
number  of  terms,  how  do  you  find  the  last  term  ? 

308.  When  you  know  the  extremes  and  the  number  of  terms,  how  do 
you  find  the  common  difference  ? 


ARITHMETICAL   PROGRESSION.  293 

EXAMPLES. 

1.  A  man  has  8  sons,  the  youngest  is  4  years  old  and  the 
eldest  32  :  their  ages  increase  in  arithmetical  progression  : 
what  is  the  common  difference  of  their  ages  ? 

2.  A  man  is  to  travel  from  New  York  to  a  certain  place  in 
12  days  ;  to  go  3  miles  the  first  day,  increasing  every  day  by 
the  same  number  of  miles,;  the  last  day's  journey  is  58  miles  : 
required  the  daily  increase. 

3.  A  man  hired  a  workman  for  a  month  of  26  working 
days,  and  agreed  to  pay  him  50  cents  for  the  first  day,  with 
a  uniform  daily  increase  ;  on  the  last  day  he  paid  $1.50  : 
what  was  the  daily  increase  ? 

CASE    III. 

309.  To  find  the  sum  of  the  terms  of  an  arithmetical 
progression. 

1.  What  is  the  sum  of  the  series  whose  first  term  is  3, 
common  difference  2,  and  last  term  19  ? 
Given  scries    -     3+    5  +    1  +    9  +  11  +  13  +  15  +  17  +  19=    99 

ofTcnnshv-l    19  +  17  +  15  +  13  +  11+   9+   t+   5+  8=   99 

verted.          J 

Sura  of  both.    2'2     iJ     22     22     22     22     22     22     22  —  198 

ANALYSIS. — The  two  series  are  the  same  ;  hence,  their  sum  is 
equal  to  twice  the  given  series.  But  their  sum  is  equal  to  the 
sum  of  the  two  extremes  3  and  19  taken  as  many  times  as  there 
are  terms  ;  and  the  given  series  is  equal  to  half  this  sum,  or  to 
the  sum  of  the  extremes  multiplied  by  half  the  number  of  terms. 

RULE. — Add  the  extremes  together  and  multiply  their 
sum  by  half  the  number  of  terms  ;  the  product  will  be  the 
sum  of  the  series. 

EXAMPLES. 

1.  The  extremes  are  2  and  100,  and  the  number  of  terms 
22  :  what  is  the  sum  of  the  series? 

OPERATION. 

ANALYSIS.— We  first  add  2  1st  term, 

together  the  two  extremes        inn   lost  tpvm 
and  then  multiply  by  half       —  la 
the  number  of  terms.  1 02  sum  of  extremes. 

11  half  the  number  of  terms 

1122  sum  of  series. 
309.  How  do  you  find  the  sum  of  the  terms? 


294  GEOMETRICAL   PKOGEESSION. 

2.  How  many  strokes  does  the  hammer  of  a  clock  strike  iu 
12  hours? 

3.  The  first  term  of  a  series  is  2,  the  common  difference  4, 
end  the  number  of  terms  9  :  what  is  the  last  term  and  sum  of 
the  series  ? 

4.  James,  a  smart  chap,  having  learned  arithmetical  pro- 
gression, told  his  father  that  he  would  chop  a  load  of  wood  of 
15  logs,  at  2  cents  for  the  first  log,  with  a  regular  increase  of 
1  cent  for  each  additional  log  :  how  much  did  James  receive 
for  chopping  the  wood  ? 

5.  An  invalid  wishes  to  gain  strength  by  regular  and  in- 
creasing exercise  ;    his  physician  assures  him  that  he  can 
walk  1  mile  the  first  day,  and  increase  the  distance  half  a 
mile  for  each  of  the  24  following  days  :    how  far  will  he 
walk  ? 

C.  If  100  eggs  are  placed  in  a  right  line,  exactly  one  yard 
from  each  other,  and  the  first  one  yard  from  a  basket :  what 
distance  will  a  man  travel  who  gathers  them  up  singlv  and 
places  them  in  the  basket  ? 


GEOMETRICAL  PROGRESSION. 

310.  A  GEOMETRICAL  PROGRESSION   is  a  series  of  terms, 
each  of  which  is  derived  from  the  preceding  one,  by  multi- 
plying it  by  a  constant  number.     The  constant  multiplier  is 
called  the  ratio  of  the  progression. 

311.  If  the  ratio  is  greater  than  1,  each  term  is  greater 
than   the   preceding   one,  and   the   series  is  said  to  be  in- 
creasing. 


31.0.  What  is  a  geometrical  progression?      What    is   the    constant 
multiplier  called  ? 

311.  If  the  ratio  is  greater  than  1,  how  do  the  terms  compare  with 
each  other?     What  is  the  series  then  called?      If   the  ratio  is  less 
than  1,  how  do  they  compare  ?    What  is  the  series  then  called  ?    What 
arc  the  several  numbers  called?    What  are  the  first  and  last  called? 
What  are  the  intermediate  ones  called  ? 

312.  How  many  parts  are  there  in  every  geometrical  progression  ? 
What  are  they?    How  manv  must  be  known  before  the  others  can  be 
found  ? 


GEOMETRICAL   PROGRESSION.  295 

If  the  ratio  is  less  than  1,  each  term  is  less  than  the 
preceding  one,  and  the  series  is  said  to  be  decreasing; 
thus, 

1,     2,    4,  8,   16;   32,  &c. — ratio  2 — increasing  series  : 
32,  16,  8,  4,    2,     1,   &c. — ratio  1 — decreasing  series. 

The  several  numbers  are  .called  terms  of  the  progression. 
The  first  and  last  are  called  the  extremes,  and  the  intermedi- 
ate terms  are  called  means. 

312.  In  every  Geometrical,  as  well  as  in  every  Arithmeti- 
cal  Progression,  there  are  five  parts  : 

1st,  the  first  term  ; 
2d,    the  last  term  ; 
3d,    the  common  ratio  ? 
4th,  the  number  of  terms  ; 
5th,  the  sum  of  all  the  terms. 

If  any  three  of  these  parts  are  known,  or  given,  the  re- 
maining ones  can  be  determined. 

• 

CASE    I. 

313.  Having  given  the  first   term,   the  ratio,    and  the 
number  of  terms,  to  find  the  last  term. 

1.  The  first  term  is  3  and  the  ratio  2  :  what  is  the  6th 
term? 

ANALYSIS. — The    se-  OPERATION. 

cond  term  is  formed  by     2x2x2x2x  2=:25  =  32 
multiplying      the     first  3   j  t  t 

term  by  the   ratio ;   tho  _____ 

third  term  by  multiply-  Ans.   96 

ing  the  second  term  by 

the  ratio,  and  so  on ;   the  number  of  multiplications  being  1  less 
iJian  the  number  of  terms  :  thus, 

3  —  3  1st  term, 

3x2  =  6  2d   term,  * 

3x2x2=3x2-=12  3d   term, 

3  x  2  x  2  x  2^3  x  23— 24  4th  term,  <fcc. 


296  GEOMETRICAL  PROGRESSION. 

Therefore,  the  last  term  is  equal  to  the  first  term  multi- 
plied by  the  ratio  raised  to  a  power  1  less  than  the  number 
of  terms. 

RULE. — Eaise  the  ratio  to  a  power  whose  exponent  is  1 
less  than  the  number  of  terms,  and  then  multiply  this  power 
by  the  first  term. 

EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  192  ;  the 
ratio  i,  and  the  number  of  terms  7  :  what  is  the  last  term  ? 

NOTE. — The  6th  power  of  the  ratio,  (•£-),  is  OPERATION. 

^4,  and  this  multiplied  by  the  first  term  192,  (l)6  =  Jk- 

gives  the  last  term  3.  1 92  X  -^-=3 

2.  A  man  purchased  12  pears  ;  he  was  to  pay  1  farthing 
for  the  1st,  2  farthings  for  the  2d,  4  for  the  3d,  and  so  on, 
doubling  each  time  :  what  did  he  pay  for  the  last  ? 

3.  The  first  term  of  a  decreasing  progression  is  1024,  the 
ratio  i :  what  is  the  9th  term  ? 

4.  The  first  term  of  an  increasing  progression  is  4,  and  the 
common  ratio  3  :  what  is  the  10th  term  ? 

5.  A  gentleman  dying  left  nine  sons,  and  bequeathed  his 
estate  in  the  following  manner  :  to  his  executors  $50  ;  his 
youngest  son  to  have  twice  as  much  as  the  executors,  and 
each  son  to  have  double  the  amount  of  the  son  next  younger : 
what  was  the  eldest  son's  portion  ? 

6.  A  man  bought  12  yards  of  cloth,  giving  3  cents  for  the 
1st  yard,  6  for  the  2d,  12  for  the  3d,  &c.  :  what  did  he  pay 
for  the  last  yard  ? 

CASE    II. 

314.  Knowing  the  two  extremes  and  the  ratio,  to  find 
the  sum  of  the  terms. 

1.  What  is  the  sum  of  the  terms  in  the  progression,  1,  4, 
16,  64  ? 

313.  Knowing  the  first  term,  the  ratio,  :ind  the  number-  of  terms,  1  row- 
do  you  find  the  Itust  term  ? 

314.  Knowing  the  two  extremes  and  the  ratio,  how  do  you  find  the 
sum  of  the  terms  V 


GEOMETRICAL   PROGRESSION.  297 

ANALYSIS. — If  we  multiply  the  terms  of  the  progression  by  the 
Tatio  4,  we  have  a  second  pro- 
gression, 4,  16,  64,  256,  which  OPERATION. 
is  4  times  as  great  as  the  first.            4+16+64+256=       4  times. 

If  from  this  we  subtract  the       1+4+16+64      =_    once. 

first,  the  remainder,  256—1,  256— 1=3  times. 

will  be  3  times  as  great  as  9~/»     1     9~,- 

the  first;  and  it  the  remain-  !—==-- =  85  sum. 

der  be  divided  by  3,  the  quo- ' 

tient  will  be  the  sum  of  the 

terms  of  the  first  progression.     But  256  is  the  product  of  the  last 

term  of  the  given  progression  multiplied  by  the  ratio,  1  is  the  first 

term,  and  the  divisor  3  is  1  less  than  the  ratio  ;  hence, 

RULE. — Multiply  the  last  term  by  the  ratio  ;  take  the  dif- 
ference between  the  product  and  the  first  term  and  divide 
the  remainder  by  the  difference  between  1  and  the  ratio. 

NOTE. — When  the  progression  is  increasing,  the  first  term  is 
subtracted  from  the  product  of  the  last  term  by  the  ratio,  and  the 
divisor  is  found  by  subtracting  1  from  the  ratio.  When  the  pro- 
gression is  decreasing,  the  product  of  the  last  term  by  the  ratio  is 
subtracted  from  the  'first  term,  and  the  ratio  is  subtracted  from  1. 


EXAMPLES. 

1.  The  first  term  of  a  progression  is  2,  the  ratio  3,  ami  the 
last  term  4374  :  what  is  the  sum  of  the  terms  ? 

2.  The  first  term  of  a  progression  is  128,  the  ratio  J,  and 
the  last  term  2  :  what  is  the  sum  of  the  terms  ? 

3.  The  first  term  is  3,  the  ratio  2,  and  the  last  term  192  : 
what  is  the  sum  of  the  series  ? 

4.  A  gentleman  gave  his  daughter  in  marriage  on  New 
Year's  day,  and  gave  her  husband  Is.  towards  her  portion, 
and  was  to  double  it  on  the  first  day  of  every  month  during 
the  year  :  what  was  her  portion  ? 

5.  A  man  bought  10  bushels  of  wheat*on  the  condition  that 
he  should  pay  1  cent  for  the  1st  bushel,  3  for  the  2d,  9  for 
the  3d,  and  so  on  to  the  last :  what  did  he  pay  for  the  last 
bushel,  and  for  the  10  bushels? 

6.  A  man  has  6  children  :  to  the  1st  he  gives  $150,  to  the 
2d  $300,  to  the  3d  $600,  and  so  on,  to  each  twice  as  much 
as  the  last :  how  much  did  the  ehKst,  Teceive,  and  what  was 
the  amount  received  by  them  all  ? 


298  PROMISCUOUS   QUESTIONS. 


PROMISCUOUS    EXAMPLES. 

1.  A  merchant  bought  13  packages  of  goods,  for  which  he  paid 
$326 :  what  will  39  packages  cost  at  the  same  rate  ? 

2.  How  many  bushels  of  oats  at  62^   cents  a  bushel  will  pay 
for  4250  feet  of  lumber  at  $7.50  per  thousand  ? 

3.  Bougkt  27ihd.  of  sugar  which  weighed  as  follows  :    the  1st 
5cwt.  Iqr.  ISlb.,  the  2d  Gcwt.  IQlb.  :  what  did  it  cost  at  7  cents  per 
pound? 

4.  How  many  hours  between  the  4th  of  Sept.,  1854,  at  3  P.M., 
and  the  20th  day  of  ApriJ,  1855,  at  10  A.M.  ? 

5.  If  |  of  a  gallon  of  wine  cost  £  of  a  dollar,  what  will  §-  of  a 
hogshead  cost  ? 

6.  What  number  is  that  which  being  multiplied  by  \  will  pro- 
duce i? 

7.  A  tailor  had  a  piece  of  cloth  containing  24£  yards,  from  which 
he  cut  6 1  yards :  how  much  was  there  left  ? 

8.  From  |  offtake  lof^' 

9.  What  is  the  difference  between  3|  +  7|  and  4  +  2-H  ? 

10.  There  was  a  company  of  soldiers,  of  whom  \  were  on  guard, 
preparing  dinner,  and  the  remainder,   85  men,  were  drilling : 

ow  many  were  there  in  the  company  ? 

11.  The  sum  of  two  numbers  is  425,  and  their  difference  1.625: 
what  are  the  numbers  ? 

12.  The  sum  of  two  numbers  is  f,  and  their  difference  ^  :  what 
are  the  numbers  ? 

13.  The  product  of  two  numbers  is  2.26,  and  one  of  the  numbers 
is  .25  :  what  is  the  other  ? 

14.  If  the  divisor  of  a  certain  number  be  6.66§,  and  the  quo- 
tient \ ,  what  will  be  the  dividend  ? 

15.  A  person  dying,  divided  his  property  between  his  widow  and 
his  four  sons ;  to  his  widow  he  gave  $1780,  and  to  each  of  his 
sons  $1250  ;  he  had  been  25^  years  in  business,  and  had  cleared 
on  an  average  126  dollars  a  year :    how  much  had  he  when  he 
began  business  ? 

16.  A  besieged  garrison  consisting  of  360  men  was  provisioned 
for  6  months,  but  hearing  of  no  relief  at  the  end  of  five  months, 
dismissed  so  many  of  the  garrison,  that  the  remaining  provision 
lasted  5  months  :  how  many  men  were  sent  away  ? 

17.  Two  persons,  A  and  B  are  indebted  to  C  ;  A  owes  $2173, 
which  is  the  least  debt,  and  the  difference  of  the  debts  is  $371  : 
what  is  the  amount  of  their  indebtedness  ? 

18.  What  number  added  to  the  43d  part  of  4429  will  make  the 
sum  240  ? 


PROMISCUOUS  QUESTIONS.  299 

19.  How  many  planks  15  feet  long,   and  15  inches  wide,   will 
floor  a  barn  60^  feet  long,  and  33i  feet  wide? 

20.  A  person  owned  f  of  a  mine,   and  sold  f  of  his  interest  for 
$  1710  :  what  was  the  value  of  the  entire  mine  ? 

21.  A  room  30  feet  long,  and  18  feet  wide,  is  to  be  covered  with 
painted  cloth  f  of  a  yard  wide  :  how  many  yards  will  cover  it  ? 

22.  A,  B  and  C  trade  together  and  gain  $120,  which  is  to  be 
shared  according  to  each  one's  stock  ;  A  put  in  $140,  B  $300,  and 
C  $160  :  what  is  each  man's  share. 

23.  A  can  do  a  piece  of  work  in  12  days,  and  B  can  do  the  same 
work  in  18  days :  how  long  will  it  take  both,  if  they  work  together? 

24.  If  a  barrel  of  flour  will  last  one  family  7£  months,  a  second 
family  9  months,  and  a  third  ll£  months,  how  long  will  it  last  the 
"three  families  together  ? 

25.  Suppose  I  have  -,%  of  a  ship  worth  $1200 ;  what  part  have 
I  left  after  selling  |  of  $  of  my  share,  and  what  is  it  worth? 

26.  What  number  is  that  which  being  multiplied  by  §  of  f  of 
1  £,  the  product  will  be  1  ? 

27.  Divide  $420  between  three  persons,  so  that  the  second  shall 
have  f  as  much  as  the  first,  and  the  third  ^  as  much  as  the  other  two  ? 

28.  What  is   the  difference   between  twice  five  and  fifty,  and 
twice  fifty  five  ? 

29.  What  number  is  that   which  being  multiplied   by   three- 
thousandths,  the  product  will  be  2637  ? 

30.  What  is  the  difference  between  half  a  dozen  dozens  and  six 
dozen  dozens? 

31.  The  slow  or  parade  step  is  70  paces  per  minute,  at  28  inches 
each  pace  :  how  fast  is  that  per  hour  ? 

32.  A  lady  being  asked  her  age,  and  not  wishing  to  give  a  direct 
answer,  said,  "  I  have  9  children,  and  three  years  elapsed  between 
the  birth  of  each  of  them  ;  the  eldest  was  born  when  I  was  19 
years  old,  and  the  youngest  is  now  exactly  19  :"  what  was  her  age  ? 

33.  A  wall  of  700  yards  in  length  was  to  be  built  in  29  days : 
12  men  were  employed  on  it  for  11  days,  and  only  completed  220 
yards :  how  many  men  must  be  added  to  complete  the  wall  in  the 
required  time  ? 

34.  Divide  $10429.50  between  three  persons,  so   that  as  often 
as  one  gets  $4,  the  second  will  get  $6  and  the  third  $7. 

35.  A  gentleman    whose  annual   income  is   £1500,    spends   20 
guineas  a  week  ;  does  he  save,  or  run  in  debt,  and  how  much  ? 

36.  A  farmer  exchanged  70  bushels  of  rye,  at  $0.92  per  bushel, 
for  40  bushels  of  wheat,  at    $1.874/  a  bushel,    and   received  the 
balance  in  oats,  at  $0.40  per  bushel :  how  many  bushels  of  oats 
did  he  receive  ? 

37.  In  a  certain  orchard  £  of  the  trees  bear  apples,  i  of  them 
bear  peaches,  £  of  them  plums,   120  of  them  cherries,  and  80  of 
them  pears:  how  many  trees  are  there  in  the  orchard  ? 


300  PKOMISCUOUS   QUESTIONS. 

38.  A  person  being  asked  the  time,  said,  the  time  past  noon 
is  equal  to  £  of  the  time  past  midnight :  what  was  the  hour  ? 

89.  If  20  men  can  perform  a  piece  of  work  in  12  days,  how 
many  men  will  accomplish  thrice  as  much  in  one-fifth  of  the  time? 

40.  How  many  stones   2  feet  long,    1  foot  wide,    and  6  inches 
thick,  will  build  a  wall  12  yards  long,  2  yards  high,  and  4  feet 
thick  ? 

41.  Four   persons   traded   together  on   a  capital    of  $6000,    of 
which  A  put  in  £,  B  put  in  ^,  C  put  in  %,  and  D  the  rest ;  at  the 
end  of  4  years  they  had  gained  $4728  :  what  was  each  one's  share  of 
the  gain  ? 

42.  A  cistern  containing  60  gallons  of  water  has  three  unequal 
pipes  for  discharging  it ;  the  largest  will  empty  it  in  one  hour,  the 
second  in  two  hours,  and  the  third  in  three  hours  :  in  what  time 
will  the  cistern  be  emptied  if  they  run  together  ? 

43.  A  man  bought  f  of  the  capital  of  a  cotton  factory  at  par  ; 
he  retained  £  of  his  purchase,  and    sold  the  balance  for    $5000 
which  was  15  per  cent  advance  on  the  cost ;  what  was  the  whole 
capital  of  the  factory  ? 

44.  Bought  a  cow  for  $30  cash,  and  sold  her  for  $35  at  a  credit 
of  8  months  :  reckoning  the  interest  at  6  per  cent,  how  much  did 
I  gain  ? 

45.  If,  when  I  sell  cloth  for  8-?.  Qd.  per  yard,  I  gain  12  per  cent, 
what  per  cent  will  be  gained  when  it  is  sold  for  10s.  Qd  per  yard  ? 

46.  How  much  stock  at  par  value  can  be  purchased  for  $8500, 
at  8^  per  cent  premium,  £  per  cent  being  paid  to  the  broker? 

47.  Twelve  workmen,  working  12  hours  a  day,  have  made  in 
12  days,  12  pieces  of  cloth,  each  piece  75  yards  long ;  how  many 
pieces  of   the  same  stuff  would  have  been  made,  each  piece  25 
yards  long,  if  there  had  been  7  more  workmen  ? 

48.  A  person  was  born  on  the  1st  day  of  Oct.,  1801,  at  6  o'clock 
in  the  morning,  what  was  his  age  on  the  21st  of  Sept.,  1854,  at 
half-past  4  in  the  afternoon? 

49.  A,  can  do  a  piece  of  work  alone  in  10  days,  and  B  in  13 
days  :  in  what  time  can  they  do  it  if  they  work  together? 

50.  A  man  went  to    sea  at  17  years  of  age;  8  years  after  he 
had  a  son  born,  who  lived  46  years,  and  died  before  his  father ; 
after  which  the  father  lived  twice  twenty  years  and  died  :  what 
was  the  age  of  the  father  ? 

51.  How  many  bricks,    8  inches  long  and  4  inches  wide,  will 
pave  a  yard  that  is  100  feet  by  50  feet  ? 

52.  If  a  house  is  50  feet  wide,  and  the  post  which  supports  the 
ridge  pole  is  12  feet  high,  what  will  be  the  length  of  the  rafters? 

53.  A  man  had  12  sons,  the  youngest  was  3  years  old  and  the 
eldest  58,  and  their  ages  increased   in  Arithmetical  progression: 
what  was  the  common  difference  of  their  ages  ? 


PROMISCUOUS   QUESTIONS.  301 

54.  If  a  quantity  of  provisions   serves  1500  men  12    weeks,  at 
the  rate  of  20  ounces  a  day  for  each  man,  how  many  men  will  the 
same  provisions  maintain  for  20  weeks,  at  the  rate  of  8  ounces  a 
day  for  each  man  ? 

55.  A  man  bought  10  bushels  of  wheat,  on  the  condition  that 
he  should  pay  1  cent  for  the  1st  bushel,  3  for  the  3d,  9  for  the  3d, 
and  so  on  to  the  last :  what  did  he  pay  for  the  last  bushel,  and  for 
the  10  bushels  ? 

56.  There  is  a  mixture  made  of  wheat  at  4s.  per  bushel,  rye  at 
3s.,  barley  at  2s.,  with  12  bushels  of  oats  at  18d.  per  bushel :  how 
much  must  be  taken  of  each  sort  to  make  the  mixture  worth  2s., 
tid.  per  bushel  ? 

57.  What  length  must  be  cut  off  a  board  8^  inches  broad   to 
contain  a  square  foot  ? 

58.  What  is  the  difference  between  the  interest  of  $2500  for  4 
years  9  mo.  at  6  per  cent.,  and  half  that  sum  for  twice  the  time, 
at  half  the  same  rate  per  cent  ? 

59.  A  person  lent  a  certain  sum  at  4  per  cent,  per  annum  ;  had 
this  remained  at  intera  Bt  3  years,  he  would  have  received  for  prin- 
cipal and  interest  $9676.80  :  what  was  the  principal? 

60.  If:  1  pound  of  tea  be  equal  in  value  to  50  oranges,  and  70 
oranges  be  worth  84  lemons,  what  is  the  value  of  a  pound  of  tea, 
when  a  lemon  is  worth  2  cents  ? 

61.  A  person  bought    160   oranges  at   2  for  a  penny,  and  180 
more  at  3  for  a  penny  ;  after  which  he  sold  them  out  at  the  rate 
of  5  for  2  pence   .did  he  make  or  lose,  and  how  much  ? 

62.  A  snail  in  getting  up  a  pole  20  feet  high,  was  observed  to 
climb  up  8  feet  every  day,  but  to  descend  4  feet  every  night :  in 
what  time  did  he  reach  the  top  of  the  pole  ? 

63.  A  ship  has  a  leak  by  which  it  would  fill  and  sink  in  15 
hours,  but  by  means  of  a  pump  it  could  be  emptied,  if  full,  in 
16  hours.     Now,  if.  the  pump  is  worked  from  the  time  the  leak 
begins,  how  long  before  the  ship  will  sink  ? 

64.  A  and  B  can  perform  a  certain  piece  of  work  in  6  days,  B 
and  C  in  7  days,   and  A  and  C  in  14  days  :  in  what  time  would 
each  do  it  alone  ? 

65.  Divide  $500  among  4  persons,  so  that  when  A  has  i  dollar 
B  shall  have  £,  C,  |,  and  D  £. 

66.  A  man  purchased    a  building  lot   containing  3600   square 
feet,  at  the  cost  of  $1.50  per  foot,  on  which  he  built  a  store  at  an 
expense  of  $3000.     He  paid  yearly  $180.66  for  repairs  and  taxes  : 
what  annual    rent  must  he  receive  to  obtain   10  per  cent  on  the 
cost? 

67.  A's  note  of  $7851.04  was  dated  Sept.  5th,  1837,  on  which 
were  endorsed    the  following  payments,    viz.  :  Nov.   13th,  1839, 
$416.98;  May  10th,  1840,  $152-  what  was  due  March  1st,  1841, 
the  interest  being  6  per  cent  ? 


302  PROMISCUOUS   QUESTIONS. 

68.  A  Louse  is  40  feet  from  the  ground  to  the  caves,  and  it  is 
required  to  find  the  length  of  a  ladder  which  will  reach  the  eaves, 
supposing  the  foot  of  the  ladder  cannot  be  placed  nearer  to  the 
house  than  30  feet  ? 

G9.  Sound  travels  about  1142  feet  in  a  second  ;  now,  if  the 
flash  of  a  cannon  be  seen  at  the  moment  it  is  fired,  and  the  report 
heard  45  seconds  after,  what  distance  would  the  observer  be  from 
the  gun  ? 

70.  A  person  dying,  worth  $5460,  left  a  wife  and  2  children,  a 
son  and  daughter,  absent  in  a  foreign  country.     He  directed  that 

"  if  his  son  returned,  the  mother  should  have  one  third  of  the  estate 
and  the  son  the  remainder ;  but  if  the  daughter  returned,  she 
should  have  one  third,  and  the  mother  the  remainder.  Now  it  so 
happened  that  they  both  returned :  how  mustthe  estate  be  divided 
to  fulfill  the  father's  intentions  ? 

71.  Two  persons  depart  from  the   same  place,  one  travels  82, 
and  the  other  36  miles  a  day  :  if  they  travel  in  the  same  direction, 
how  far  will  they  be  apart  at  the  end  of  19  days,  and  how  far  if 
they  travel  in  contrary  directions  ? 

72.  In  what  time  will  $2377.50  amount  to  $2852.42,  at  4  per 
cent,  per  annum  ? 

73.  What  is  the  height  of  a  wall,  which  is  14^  yards  in  length, 
and  -fo  of  a  yard  in  thickness,  and  which  has  cost  $406,  it  having 
been  paid  for  at  the  rate  of  $10  per  cubic  yard  ? 

74.  What  will  be  the  duty  on  225  bags  of  coffee,  each  weighing 
gross  160  Ibs.,  invoiced  at  6  cents  per  Ib.  ;  2  per  cent,  being  the 
legal  rate  of  tare,  and  20  per  cent,  the  duty  ? 

75.  Three  persons  purchase  a  piece  of  property  for  $9202  ;  the 
first  gave  a  certain   Bum ;  the  second   three  times  as  much  ;  and 
the  third  one  and  a  half  time  as  much  as  the  other  two:  what 
did  each  pay  ? 

76.  A  reservoir  of  water  has  two  pipes  to  supply  it.     The  first 
would  fill  it  in  40  minutes,  and  the  second  in  50.     It  has  likewise 
a  discharging  pipe,  by  which  it  may  be  emptied  when  full  in  25 
minutes.    Now,  if  all  the  pipes  are  opened  at  once,  and  the  water 
runs  uniformly  as  we  have  supposed,  how  long  before  the  cistern 
will  be  filled? 

77.  A  traveller    leaves  New  Haven   at  8  o'clock   on    Monday 
morning,  and  walks  towards  Albany  at  the   rate  of  3  miles  an 
hour :  another  traveller  sets  out  from  Albany  at  4  o'clock  on  the 
same  evening,   and  walks  towards   New  Haven  at  the  rate  of  4 
miles  an   hour  ;    now,   supposing  the  distance  to  be  130  miles, 
where  on  the  road  will  they  meet  ? 


MENSURATION.  303 


MENSURATION. 

315.  A  triangle  is  a  portion  of  a  plane 
bounded  by  three  straight  lines.    BC  is 
called  the  base  ;  and  AD,  perpendicular  to 
BC,  the  altitude. 

316.  To  find  the  area  of  a  triangle. 
The  area  or  contents  of  a  triangle  is  equal 

to  the  product  of  half  its  base  by  its  altitude 
(Bk.  IV.  Prop.  VI).* 

EXAMPLES. 

1.  The  base,  BC,  of  a  triangle  is  40  yards,  and  the  perpendicu- 
lar, AD,  20  yards ;  what  is  the  area  ? 

2.  In  a  triangular  field  the  base  is  40  chains,  and  the  perpendi- 
cular 15  chains :  how  much  does  it  contain  ?     (ART.  110.) 

3.  There  is  a  triangular  field,  of  which  the  base  is  35  rods  and 
the  perpendicular  26  rods :  what  are  its  contents  ? 


317.  A?  square  is  a  figure  having  four  equal  sides, 
and  all  its  angles  right  angles. 


318.  A  rectangle  is  a  four-sided  figure  like  a 
square,  in  which  the  sides  are  perpendicular  to  each 
other,  but  the  adjacent  sides  are  not  equal. 

319.  A    parallelogram  is  a  four-sided  figure 
which  has  its  opposite  sides  equal  and  parallel,  but 
its  angles  not  right  angles.  The  line  DE,  perpendi- 
cular to  the  base,  is  called  the  altitude. 


320.  To  find  the  area  of  a  square,  rectangle,  or  parallelogram, 


Multiply  the  base  by  the  perpendicular  height,  and  the  product 
will  be  the  area.    (Book  IV.  Prop.  V). 

EXAMPLES. 

1.  What  is  the  area  of  a  square  field  of  which  the  sides  are 
each  33.08  chains  ? 

2.  What  is  the  area  of  a  square  piece  of  land  of  which  the 
sides  are  27  chains? 

3.  What  is  the  area  of  a  square  piece  of  land  of  which  the  sides 
are  25  rods  each  ? 

*  All  the  references  arc  to  Davies'  Legendre. 


304:  MENSURATION. 

4.  What  are  the  contents  of  a  rectangular  field,  the  length  of 
which  is  40  rods  and  the  breadth  20  rods  ? 

5.  What  are  the  contents  of  a  field  40  rods  square  ? 

6.  What  are  the  contents  of  a  rectangular  field  15  chains  long 
and  5  chains  broad  ? 

7.  What  are  the  contents  of  a  field  27  chains  long  and  9  rods 
broad  ? 

8.  The  base  of  a  parallelogram  is  271  yards,  and  the  perpendi. 
cular  height  360  feet :  what  is  the  area  ? 

321.  A    trapezoid    is    a    four-sided    figure 
ABCD,   having    two    of   its    sides,   AB,    DC, 
parallel.      The    perpendicular    CE    is    called 
the  altitude. 

322.  To  find  the  area  of  a  trapezoid. 

Multiply  half  the  sum  of  the  two  parallel  sides  "by  the  alti- 
tude, and  the  product  will  be  the  area.  (Bk.  IV.  Prop.  VII.) 

EXAMPLES. 

1.  Required  the  area  of  the  trapezoid  ABCD,  having  given 

AB=321.51/£.,    DC=214.24/*.,     and  CE=171.16/^. 

2.  What  is  the  area  of  a  trapezoid,  the  parallel  sides  of  which 
are  12.41  and  8.22  chains,  and  the  perpendicular  distance  between 
them  5.15  chains '? 

3.  Required  the  area  of  a  trapezoid  whose  parallel  sides  are  25 
feet  6  inches,  and  18  feet  9  inches,  and  the  perpendicular  distance 
between  them  10  feet  and  5  inches. 

4.  Required  the  area  of  a  trapezoid  whose  parallel  sides  are 
20.5   and    12.25,  and   the   perpendicular   distance   between   them 
10.75  yards. 

5.  What  is  the  area  of  a  trapezoid  whose  parallel  sides  are  7.50 
chains,  and  12.25  chains,  and  the  perpendicular  height  15.40  chains  V 

6.  What  are  the  contents  when  the  parallel  sides  are  20  and  32 
chains,  and  the  perpendicular  distance  between  them  26  chains  ? 

323.  A  circle    is   a   portion    of    a    plane 
bounded  by  a  curved  line,  called  the  circum- 
ference.    Every  point  of  the  circumference  is 
equally  distant   from  a   certain   point  within 
called  the  centre  :   thus,  C  is  the  centre,  and 
any  line,  as  ACB,  passing  through  the  centre, 
is  called  a  diameter. 

If  the  diameter  of  a  circle'  is  1,  the  circumference  will  be 
3.1416.  Hence,  if  we  know  the  diameter,  ^ce  may  find  the  circum- 
ference by  multiplying  by  3.1416  ;  or,  if  we  know  the  circumference., 
we  may  find  the  diameter  by  dividing  by  3.1416. 


MENSURATION.  305 

EXAMPLES. 

1.  The  diameter  of  a  circle  is  4,  what  is  the  circumference  ? 

2.  The  diameter  of  a  circle  is  93,  what  is  the  circumference  ? 

3.  The  diameter  of  a  circle  is  20,  what  is  the  circumference  ? 

4.  What  is  diameter  of  a  circle  whose  circumference  is  78.54  ? 

5    What  is  the    diameter    of  a   circte  whose   circumference  is 
11052.1944? 
6.  What  is  the  diameter  of  a  circle  whose  circumference  is  6850  ? 

324.  To  find  the  area  or  contents  of  a  circle. 

Multiply  the  square  of  the  diameter  by  the  decimal  .7854  (Bk.  V. 
Prop.  XII.  Cor.  2). 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  6  ? 

2.  What  is  the  area  of  a  circle  whose  diameter  is  10? 

3.  What  is  the  area  of  a  circle  whose  diameter  is  7  ? 

4.  How  many  square  yards  in  a  circle  whose  diameter  is  3i  feet  ? 

325.  A  sphere  is  a  figure  terminated 
by  a  curved  surface,  ull  the  parts  of  which 
are  equally  distant  from  a  certain  point 
within  called  the  centre.     The  line  AB 
passing  through  its  centre  C  is  called  the 
diameter  of  the  sphere,  and  AC  its  radius. 

o~6.   To  find  the  surface  of  a  sphere, 

Multiply  the  square  of  the  diameter  by 
3.1416  (Bk.  VIII.  Prop.  X.  Cor). 

EXAMPLES. 

1.  What  is  the  surface  of  a  sphere  whose  diameter  is  12  ? 

2.  What  is  the  surface  of  a  sphere  whose  diameter  is  7  ? 

3.  Required  the    number  of   square  inches  in  the  surface  of  a 
sphere  whose  diameter  is  2  feet  or  24  inches. 

327.   To  find  the  contents  of  a  sphere, 

Multiply  the  surface  by  the  diameter  and  divide  the  product  by  6; 
the  quotient  mil  be  the  contents.  (Bk.  VIII.  Prop.  XIV.  Sch.  3.) 

EXAMPLES 

1.  What  are  the  contents  of  a  sphere  whose  diameter  is  12  ? 

2.  What  are  the  contents  of  a  sphere  whose  diameter  is  4  ? 

3.  What  are  the  contents  of  a  sphere  whose  diameter  is  14i7i.  ? 
4  What  are  the  contents  of  a  sphere  whose  diameter  is  Gfl.  ? 

20 


306 


MENSURATION. 


328.  A  prism  is  a  figure  whose  ends  are  equal 
plane  figures  and  whose  faces  are  paralelograms. 

The  sum  of  the  sides  which  bound  the  base  is 
called  the  perimeter  of  the  base,  and  the  sum  of  the 
parallelograms  which  bound  the  solid  is  called  the 
convex  surface. 


329.  To  find  the  convex  surface  of  a  right  prism, 

Multiply  the  perimeter  of  the  base  by  the  perpendicular  height,  and 
thegtroduct  will  be  the  convex  surface  (Bk.  VII.  Prop.  I). 

EXAMPLES. 

1.  What  is  the  convex  surface  of  a  prism  whose  base  is  bounded 
by  five  equal  sides,  each  of  which  is  35  feet,  the  altitude  being  26 
feet? 

2.  What  is  the  convex  surface  when  there  are  eight  equal  sides, 
each  15  feet  in  length,  and  the  altitude  is  12  feet  ? 

330.  To  find  the  solid  contents  of  a  prism. 

Multiply  the  area  of  the  base  by  the  altitude,  and  the  product  will 
be  the  contents  (Bk.  VII.  Prop.  XIV). 

EXAMPLES. 

1.  What  are  the  contents  of  a  square  prism,  each  side  of  the 
square  which  forms  the   base  being  15,  and  the  altitude  of  the 
prism  20  feet  ? 

2.  What  are  the  contents  of  a  cube  each  side  of  which  is  24 
inches  ? 

3.  How  many  cubic   feet  in  a  block   of  marble    of  which   the 
length  is  3  feet  2  inches,  breadth  2  feet  8  inches  and  height  or 
thickness  2  feet  6  inches  ? 

4.  How  many  gallons  of  water  will  a  cistern  contain  whose  di- 
mensions are  the  same  as  in  the  last  example  ? 

5.  Required  the  contents  of  a  triangular  prism  whose  height  is 
10  feet,  and  area  of  the  base  350  ? 


331.  A  cylinder  is  a  figure  with  circular 
ends.  The  line  EF  is  called  the  axis  or  alti- 
tude, and  the  circular  surface  the  convex  sur- 
face of  the  cylinder. 


MENSURATION. 


307 


332.  To  find  the  convex  surface, 

Multiply  the  circumference  of  the  base  by  the  altitude,  and 
the  product  ivill  be  the  convex  surface.   (Bk.  VIII.  Prop.  I.) 

EXAMPLES. 

1    What  is  the  convex  surface  of  a  cylinder,  the  diameter  of 
whose  base  is  20  and  the  altitude  50  ? 

2.  What  is  the  convex  surfa'ce  of  a  cylinder,  whose  altitude  is 
14  feet  and  the  circumference  of  its  base  8  feet  4  inches  ? 

3.  What  is  the  convex  surface  of  a  cylinder,  the  diameter  of 
whose  base  is  30  inches  and  altitude  5  feet  ? 

333.  To  find  the  contents  of  a  cylinder, 

Multiply  the  area  of  the  base  by  the  altitude  :  the  product  will  be 
the  contents.    (Bk.  VIII.  Prop.  II). 


EXAMPLES. 

1.  Required  the  contents  of  a  cylinder  of  which  the  altitude  is 
12  feet  and  the  diameter  of  the  base  15  feet  ? 

2.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whoso 
base  is  20  and  the  altitude  29? 

3.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  12  and  the  altitude  30  ? 

4.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  16  and  altitude  9  ? 

5.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  50  and  altitude  15  ? 


334.  A  pyramid  is  a  figure  formed  by 
several  triangular  planes  united  at  the 
same  point  S,  and  terminating  in  the 
different  sides  of  a  plain  figure  as 
ABCDE.  The  altitude  of  the  pyramid 
is  the  line  SO,  drawn  perpendicular  to 
the  base. 


335.  To  find  the  contents  of  a  pyramid, 

Multiply  the  area  of  the  base  by  one-third  of  the  altitude. 
(Bk.  VII,  Prop  XVII). 


308 


MENSUEATlQJS'. 


EXAMPLES. 

1.  Required  the  contents  of  a  pyramid,  of  which  the  area  of  the 
base  is  95  and  the  altitude  15. 

2  What  are  the  contents  of  a  pyramid,  the  area  of  whose  base 
is  260  and  the  altitude  24  ? 

3.  What  are  the  contents  of  a  pyramid,  the  area  of  whose  base 
is  207  and  altitude  18? 

4  What  are  the  contents  of  a  pyramid,  the  area  of  whose  base 
is  403  and  altitude  30  ? 

5.  What  are  the  contents  of  a  pyramid,  the  area  of  whose  base 
is  270  and  altitude  16? 

6.  A  pyramid  has  a  rectangular  base,  the  sides  of  which  are  25 
and  12 :    the  altitude  of  the  pyramid  is  36  :   what   are   its  con- 
tents ? 

7.  A  pyramid  with  a  square  base,  of  which  each  side  is  30,  has 
an  altitude  of  20  :  what  are  its  contents  ? 


336.  A  cone  is  a  figure  with  a  circular 
base,  and  tapering  to  a  point  called  the 
vertex.  The  point  C  is  the  vertex,  and  the 
line  CD  is  called  the  axis  or  altitude. 


337.  To  find  the  contents  of  a  cone, 

Multiply  the  area   of  the  base  ly  one-third   of  tJie  altitude. 
(Bk.  VIII.  Prop.  V.) 

EXAMPLES. 

1.  Required  the  contents  of  a  cone,  the  diameter  of  whose  base 
is  5  and  the  altitude  10. 

2.  What  are  the  contents  of  a  cone,  the  diameter  of  whose  base 
is  18  and  the  altitude  27  ? 

3.  What  are  the  contents  of  a  cone,  the  diameter  of  whose  base 
is  20  and  the  altitude  30  ? 

4.  What  are  the  contents  of  a  cone,  whose  altitude  is  27  feet 
and  the  diameter  of  the  base  10  feet  ? 

5.  What  are  the  contents  of  a  cone,  whose  altitude  is  12  feet 
and  the  diameter  of  its  base  15  feet  ? 


GAUGING  309 

GAUGING-. 

338.  The  mean  diameter  of  a  cask  is  found  by  adding  to  tho 
head  diameter,  two  thirds  of  the  difference  between  the  bung  and 
head  diameters,  or  if  the  staves  are  not  much  curved,  by  adding" 
six-tenths.     This  reduces  the  cask  to  a  cylinder.     Then,  to  find 
the  solidity,  we  multiply  the  square  of  the  mean  diameter  by  the 
decimal  .7854  and    the   product    by  the   length.     This  will   give 
the  solid  contents  in  cubic  inches.     Then,  if  we  divide  by  231, 
we  have  the  contents  in  gallons.     (Art.  114). 

Multiply  the  length  by  the  square  of  the         OPERATION. 
mean   diameter,  then   by  the  decimal  .7854,       Ixd2  X  '-7-8 w5-^- — 
and  divide  by  231.  '  J  x  d2  x  .0034. 

If,  then,  we  divide  the  decimal  ,7854  by  231,  the  quotient  car- 
ried to  four  places  of  decimals  is  .0034,  and  this  decimal  multi- 
plied by  the  square  of  the  mean  diameter  and  by  the  length  of  the 
cask,  will  give  the  contents  in  gallons. 

339.  Hence,  for  gauging  or  measuring  casks,  we  have  the  fol. 
lowing 

HULE. — Multiply  the  length  "by  the  square  of  the  mean  diameter  ; 
then  multiply  ly  34  and  point  off  four  decimal  places,  and  the  pro- 
duct icill  then  express  gallons  and  the  decimals  of  a  gallon. 

1.  How  many  gallons  in  a  cask  whose  bung  diameter  is  36 
inches,  head  diameter-  30  inches,  and  length  50  inches  ? 

We  first  find  the  difference  of  the  diameters,  OPERATION. 

of  which  we  take  two  thirds  and  add  to  the  36—30=    6 

head  diameter.    We  then  multiply  the  square  2  Of  6   =   4 

of  the   mean    diameter,  the   length   and  34  3Q()4-4.—  °4 
together,  and  point  off  four  decimal  places 

in  the  product.  34  =1156 

2.  What  is   the    number   of  gallons  in  a          -.  Qr  r  0     7 
cask  whose  bung  diameter  is  38  inches,  head  lyo.O^roc. 
diameter  32  inches,  and  length  42  inches  ? 

3.  How  many  gallons  in  a  cask  whose  length  is  36  inches,  bung 
diameter  35  inches,  and  head  diameter  30  inches  ? 

4.  How  many  gallons  in  a  cask  whose  length  is  40  inches,  head 
diameter  34  inches,  and  bung  diameter  38  inches? 

5  A  water  tub  holds  147  gallons  ;  the  pipe  usually  brings  in 
14  gallons  in  9  minutes  :  the  tap  discharges  at  a  medium,  40  gal- 
Jons  in  31  minutes.  Now,  supposing  these  to  be  left  open,  and 
the  water  to  be  turned  on  at  2  o'clock  in  the  morning ;  a  servant 
at  5  shuts  the  tap,  and  is  solicitous  to  know  at  what  time  the  tub 
will  be  filled  in  case  the  water  continues  to  flow. 


310 


APPENDIX, 


FORMS  RELATING  TO  BUSINESS  IN  GENERAL. 


FORMS   OF   OKDERS. 

MESSRS.  M.  JAMES  &  Co. 

Please  pay  John  Thompson,  or  order,  five  hundred 
dollars,  and  place  the  same  to  my  account,  for  value  received. 

PETER  WORTHY. 
Wilmington,  N.  0.,  June  1,  1855. 

MR.  JOSEPH  RICH, 

Please  pay,  for  value  received,  the  bearer,  sixty-one 
dollars  and  twenty  cents,  in  goods  from  your  store,  and  charge  the 
same  to  the  account  of  your 

Obedient  Servant 

JOHN  PARSONS. 
Savannafi,  Ga.,  July  1,  1855. 


FORMS   OF   RECEIPTS. 

Receipt  for  Money  on  account. 

Received,  Natchez,  June  2d,  1855,  of  John  Ward,  sixty  dollars 
on  account. 

$60,00  JOHN  P.  FAY. 

Receipt  for  Money  on  a  Note. 

Received,  Nashville,  June  5,  1856,  of  Leonard  Walsh,  six  hun- 
dred and  forty  dollars,  on  his  note  for  one  thousand  dollars,  dated 
New  York,  January  1,  1855. 

$640,00  J.  N.  WEEKS. 

NOTES. 

1.  A  NOTE,  or  as  it  is  generally  called,  a  promissory  note,  is  a 
positive  engagement,  in  writing,  to  pay  a  given  sum  at  a  time 
specified,  either  to  a  person  named  in  the  note,  or  to  his  order,  or 
to  the  bearer. 

2.  By  mercantile  usage  a  note  does  not  really  fall  due  until  the 
expiration  of  3  days  after  the  time  mentioned  on  its  face.     The 
three  additional  days  are  called  days  of  grace. 


APPENDIX.  311 

When  the  last  day  of  grace  happens  to  be  Sunday,  or  a  holiday, 
such  as  New  Years,  or  the  Fourth  of  July,  the  note  must  be  paid 
the  day  before  :  that  is,  on  the  second  day  of  grace. 

3.  There  are  two  kinds  of  notes  discounted  at  banks  :  1st.  Notes 
given  by  one  individual  to  another  for  property  actually  sold— 
these  are  called  business  notes,  or  business  paper.  3d.  Notes  made 
for  the  purpose  of  borrowing  money,  which  are  called  accommo- 
dation notes,  or  accommodation  paper.  Notes  of  the  first  class  are 
much  preferred  by  the  banks,  as  more  likely  to  be  paid  when  they 
fall  due,  or  in  mercantile  phrase,  "  when  they  come  to  maturity." 

FORMS  OP  NOTES. 

No.  1.  Negotiable  Note. 


$25,50.  .        Providence,  May  1, 1856. 

For  value  received  I  promise  to  pay  on  demand,  to  Abel 
Bond,  or  order,  twenty-five  dollars  and  50  cents. 

REUBEN  HOLMES. 


Note  Payable  to  Bearer. 
No.  2. 


$875,39.  St.  Louis,  May  1,  1855. 

For  value  received  I  promise  to  pay,  six  months  after 
date,  to  John  Johns,  or  bearer,  eight  hundred  and  seventy-five 
dollars  and  thirty-nine  cents. 

PIERCE  PENNY. 


Note  by  two  Persons. 
No.  3. 


$659,27.  Buffalo,  June  2,  1856. 

For  value  received  we,  jointly  and  severally,  promise  to 
pay  to  Richard  Ricks,  or  order,  on  demand  sis  hundred  and  fifty- 
nine  dollars  and  twenty-seven  cents. 

ENOS  ALLAN. 

JOHN  ALLAN. 


Note  Payable  at  a  Bank. 

$20,25.  Chicago,  May  7,  1856. 

Sixty  days  after  date,  I  promise  to  pay  John  Anderson, 
or  order,  at  the  Bank  of  Commerce  in  the  city  of  New  York, 
twenty  dollars  and  twenty-five  cents,  for  value  received. 

JESSE  STOKES. 


312  APPENDIX. 


REMARKS    RELATING   TO    NOTES. 

1.  The  person  who  signs  a  note,  is  called  the  drawer  or  maker 
of  the  note ;  thus,  Reuben  Holmes  is  the  drawer  of  Note  No.  1. 

2.  The  person  who  has   the  rightful  possession  of  a  note,  is 
called  the  holder  of  the  note. 

3.  A  note  is  said  to  be  negotiable  when  it  is  made  payable  to 
A  B,  or  order,  who  is  called  the  payee  (see  No.  1).     Now,  if  Abel 
Bond,  to  whom  this  note  is  made  payable,  writes  his  name  on  the 
back  of  it,  he  is  said  to  endorse  the  note,  and  he  is  called  the  en- 
dorser ;    and  when  the  note    becomes   due,  the   holder  must  first 
demand  payment  of  the  maker,  Reuben  Holmes,  and  if  he  declines 
paying  it,  the  holder  may  then  require  payment  of  Abel  Bond,  the 
endorser. 

4  If  the  note  is  made  payable  to  A  B,  or  bearer,  then  the 
drawer  alone  is  responsible,  and  he  must  pay  to  any  person  who 
holds  the  note. 

5.  The  time  at  which  a  note  is  to  be  paid  should   always  be 
named,  but  if  no  time  is  specified,  the  drawer  must  pay  when  re- 
quired to  do  so,  and  the  note  will  draw  interest  after  the  payment 
is  demanded. 

6.  When  a  note,  payable  at  a  future  day,  becomes  due,  and  is 
not  paid,  it  will  draw  interest,  though  no  mention  is  made  of  inter- 
est. 

7.  In  each  of  the  States  there  is  a  rate  of  interest  established  by 
law,  which  is  called  the  legal  interest,  and  when  no  rate  is  speci- 
fied, the  note  will  always  draw  legal  interest.     If  a  rate  higher 
than  legal  interest  be  taken,  the  drawer,  in  most  of  the  States,  is 
not  bound  to  pay  the  note. 

8.  In  the  State  of  New  York,  although  the  legal  interest  is  7 
per  cent,  yet  the  banks  are  not  allowed  to  charge  over  G  per  cent, 
unless  the  notes  have  over  63  days  to  run. 

9.  If  two  persons  jointly  and  severally  give  their  note,  (see  No. 
3,)  it  may  be  collected  of  either  of  them. 

10.  The  words  "For  value   received"  should   bo   expressed  in 
every  note. 

11.  When  a  note  is  given,  payable  on  a  fixed  day,  and  in  a  spe- 
cific article,  as  in  wheat  or  rye,  payment  must  be  offered  at  the 
specified  time,  and  if  it  is  not,  the  holder  can  demand  the  value  in 
money. 

A    BOND    FOR    ONE    PERSON,  WITH    A    CONDITION. 

KNOW  ALL  MEN  BY  THESE  PRESENTS,  THAT  I,  James 
Wilson  of  the  City  of  Hartford  and  State  of  Connecticut,  am  held 
and  firmly  bound  unto  John  Pickens  of  the  Town  of  Waterbury, 
County  of  New  Haven  and  State  of  Connecticut,  in  the  sum  of 


APPENDIX. 


313 


Eighty  dollars  lawful  money  of  the  United  States  of  America,  to 
be  paid  to  the  said  John  Pickens,  his  executors,  administrators,  or 
assigns  :  for  which  payment  well  and  truly  to  be  made  J  bind 
myself,  my  heirs,  executors,  and  administrators,  firmly  by  these 
presents.  Sealed  with  my  Seal.  Dated  the  Ninth  day  of  March, 
one  thousand  eight  hundred  and  thirty-eight. 

THE  CONDITION  of  the  above  obligation  is  such,  that  if  tlio 
above  bounden  James  Wilson,  his  heirs,  executors,  or  administra- 
tors, shall  well  and  truly  pay  or  cause  to  be  paid,  unto  the  above- 
named  John  Pickens,  his  executors,  administrators,  or  assigns,  the 
just  and  full  sum  of 

[Here  insert  the  condition.] 

then  the  above  obligation  to  be  void,  otherwise  to  remain  in  full 
force  and  virtue. 


Sealed  and  delivered  in 
the  presence  of 

John  Frost,         ) 
Joseph  Wiggins,) 


James  Wilson. 


NOTE.— The  part  in  Italic  to  be  filled  up  according  to  circum- 
stance. 

If  there  is  no  condition  to  the  bond,  then  all  to  be  omitted  after 
and  including  the  words,  "  THE  CONDITION,  &c." 


BOOK-KEEPING. 

PERSONS  transacting  business  find  it  necessary  to  wiite  down 
the  articles  bought  or  sold,  together  with  their  prices  and  the 
names  of  the  persons  to  whom  sold. 

BOOK-KEEPING  is  the  method  of  recording  such  transactions  in  a 
regular  manner. 

COMMON  ACCOUNT  BOOK. 

The  following  is  a  very  convenient  form  for  book-keeping,  and 
requires  but  a  single  book.  l"c  is  probably  the  best  form  of  a  com- 
mon Account  Book. 


J.  BELL.            DR                  J.  BELL.                     CR. 

1846. 

$ 

c. 

1846. 

*|«. 

June  1 

11    6 
July  9 

To  5  cords  of  wood, 
at  $1,75  per  cord, 
To  1  day's  work, 
To  4bn.  of  rye,  at  62 
cents  per  bu. 

8 
1 

2 

75 

00 

48 

July  ( 
"  1C 
u  20 
Aug.  1 

By  shoeing  horse, 
u  mending  sleigh, 
"  ironing  wagon, 
"  Cash  to  balance, 

100 

325 
512 
386 

12 

23 

12i23 

314 


ANSWERS. 


p. 

EX. 

ANS. 

EX. 

ANS 

EX 

ANS. 

EX. 

ANS. 

24. 
24. 

9 

10 

577 
7689 

11 

12 

502616 
799999 

13 
14 

43  cts. 
73  cts. 

15 

|888 

20. 
25. 

17 

18 

4083 
6846 

19 
20 

9798 
8601 

21 

22 

7032 
979 

23 

559 

2Ve 

27. 
27. 
27. 

5 
6 

7 
8 

12089 
26901 
28637 
203933 

9 
10 
11 
12 

23272   - 
233642 
247481 
1994439 

13 
14 
15 
16 

175874 
172775 
98967 
10742750 

28. 
28. 
28. 

20 
21 
22 

787676921 
100570011 
15371781930 

23 
24 
25 

26754 
730528 

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2fc 

27 

25687540 
297303078 

29. 
29. 
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28 
29 
30 

13115375 
3942805S 
140700034 

31 
32 
33 

1819857171537 
1105354 
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34 

1118969 

30.||  1 

365  ||  2 

5567  ||3|  16375||4|421||5|392||6 

34671660 

31. 
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7 
8 
9 

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11 

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J4239052< 
(  453090S 

)   12 
)   13 
> 

1287462 
1665400 

32. 
32. 
32. 

14 

15 
16 

50994 
143985 
2728116 

17 
18 
19 

5990267 
6644374 
7685134 

20 
21 

23191876 
23191876 

37. 
37. 
37. 
37. 

9 
10 
11 
12 

260822 
2935621 
50391719 
28443 

13 
14 
15 
16 

99246591 
999999 
776462 
18561747 

17 

18 
19 

4244083 
8013105 

52528 

38. 
38. 
38. 

1 
2 
3 

10   - 
45 
$1115 

4  23^ 

5 

5   6 

t 
f 

7 
8 
9 

62 

785608 
37 

10 
11 
12 

175502 
696 

2687 

39. 
39. 
39. 
39. 

13 
14 
15 
16 

250-$1500 

26 
1860805 

17 
18 
19 
20 

239 

1759 
55 

21 
22 
23 
24 

190 
$4020-1340 
2769818 
94 

ANSWERS. 


315 


p. 

EX. 

ANS. 

EX. 

ANS. 

EX. 

ANS 

EX. 

ANS. 

40. 
40. 

25 
26 

145 

168 

27 
28 

168 
137 

29 
30 

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36914176 

22 

6241519790 

49. 

5 

4280822 

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85950000 

23 

105062176 

49. 

6 

19014604 

15 

3320863272 

24 

601380780 

49. 

7 

85564584 

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816515040 

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321 


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8  1.333+.162  +  .792 
9  .85 
10  ,075  ' 

11 
12 
13 
14 

136 
00875 
2976 
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15 
16 

17 
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198. 

198. 
198. 
198. 
198. 

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3    W»- 

4      TTrooTTIr 

1 

2 
3 
4 
5 

.02734375/6 
£108333  + 
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1.3125p&. 

6 
7 
8 
9 
10 

ife 

199. 
199. 
199 
199. 
199. 
199. 
199. 
199. 
199. 

1 

2 
3 
4 

5 
6 

7 
8 
9 

12.00384(7?-. 
2<?r.  12/6.  8oz. 

%qt.  \pt. 

6s.  9d. 
&cwt.  3or. 
8P. 
Ihhd.  4^  gal.  \qt. 
6gal.  Zqt. 
136do.  2  Mr. 

10 
11 
12 
13 
14 
15 
16 
17 

Is.  8^7. 
3$r.  1] 
19/ir.  i 

loz.  8c 

£1  Os. 
£1  17. 

1* 
1/6. 
2ln 

Ir. 
237 
11 

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far. 

i.  36sec. 
.  *[fl.  11. 

ir.  59m. 

12  48s<?c. 

200. 
200. 
200. 
200. 
200. 

1 
2 
3 
4 
5 

4.889955M>&.+ 
2.4694/6.+ 
1.25yd, 

1.046875/6. 
5.0833.L.+ 

6 
i 

8 
9 
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4.8906256M 

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5.88125^. 
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1  .42859226^. 
2  .39201c7z. 
3  7.8781253/: 

203. 
203. 

6 

7 

$36.428 

$21.25 

8  $28.333  + 

9  $32.812  + 

10$30.833 

11|$62. 

1   12 
13 

$3.111  + 
24  pounds. 

204.j|15|472,50||16|6  days'  i£ork.\\ 

17;31i|6M,|jl8  |  $18,541  + 

205. 

!  20     $8. 

||  21  |  $25.50 

23 

49  men. 

||  24 

|  Uwk. 

20G.||  26  |  18 

bales.  ||  27  |  \\\ft. 

long.  ||  29 

1  2*< 

iays. 

207. 
207. 

3 

3 

n 

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2  10§  " 

3 

4    (  A's  gjn  $58.33^ 

35 

1st. 
3d. 

$240  2d.  $200 
$140 

208. 
209. 

58     1£ 
9|30do. 

days. 

— 

— 

—  - 

||40  9rfa.||41|$96||43|72  u-o'72.|j44  42 

Georgia. 

210. 
210. 
210. 
210 

210. 

11 

21 

q 

4 
5 

18  sheep. 
$112 

48da. 

$6.5625 

6 
7 
8 
9 
10 

12§6ar. 
$1.60 

$17.273-f 

11 

-    12 
13 
14 

(3U//.= 
(  lOJyaT. 
iOOda. 

$10 
lOmo. 

15 

16 
17 

1st.  $2.50 
2(7.  $3.75 
3d.  $8.75 
22i<?a/: 
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ANSWERS* 


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p. 

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211. 

211. 
211. 
211. 

EX 

ANS. 

EX 

ANS. 

EX 

ANS. 

EX        ANS. 

18 
19 

20 

$9.16§ 

(  1st.  105/6. 
12d.  140/6. 
(3d.  168/6. 

|21  i 
22 

23^ 

241 

($175 

]<£$! 
^10500 

25 

25 

26 

27 
28 
29 

$18 
$83.33| 

3  men. 

30  36  men 

[All 

QI      -515 

'  1  (724 

[  hogs. 

212.| 
212. 
212 
212'. 

32 

33 
34 

(  As  $126.  B's 

j  $117.  C's$72. 
546ar. 

35  1 
36] 

37$ 
38  S 

50.803  + 

^1344 
55040 

39 

40 
41 

1st.  $39.20   2d. 
$19.60  3d.  117.60 
$533.331 
3  pieces. 

216.||  1 

36  ||  2 

60 

II  3 

12  || 

4 

2J  II  5  |  24      — 

217. 
217. 

21 

4- 

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?XS4> 

8 
9 

Si!  ^ 

tOj-a—  li. 
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219. 

219. 
219. 

1 

2 
3 

308mi.        4    3300  pounds. 
$165           5    $61.425 
$1381.25    6     10955mi. 

7 

S 
10 

9243,7  5 
*20  1,75 

36000  rations. 

220. 
220. 
220. 
220. 
220 

11 
13 
14 
15 
16 

§Yr.  20m. 
L861  + 
227  12s.  ld.+ 
$115.50 

$29.25 

117 
18 

19 
20 

Sl^ft.                  2 
140.32                  2 
(As  $1787.50  2 
]  £'s  $1283.75  - 
122.50 

1  $1.871 
2  $0.154  + 
3  $6206.931 

221. 
221. 
221. 
221. 

24 

25 
26 
27 

$61.425 
5s.  9d. 
3156w. 

28: 
29 
30 
31 

$252 
24yd 
72  hats 
376ar. 

32 

33 
34 

35 

21f/6. 
$12.13 

28/ir. 
%%  acres 

36$] 

37$] 
38  $j 

39  $5 

.8.27 
68.742  + 
08.25 
53.125 

223.11  1  |  8  days. 

224. 

224. 
224. 

2 

3 
4 

27da.     5 
72da.     6 
160da.    7 

20/irses 
18da. 

8 
S 
1C 

27da. 

11 

12 
13 

256 
lOd 

i.     14    1-fclb. 

a., 

22(). 

1  |  $45  ||  S 

5     150/6.H  3 

$99||  4 

232da.  ||  5     511i??ii'. 

227. 
227. 
227. 
227. 

6 

7 
8 
9 

18yr. 
27  weav 
72  men. 

10   4JTda. 
11    11126a. 

's  12    2^-  tons 
13    343  1  ft. 

14 
15 
16 
17 

15/6. 
38fraz's 
1926ar. 
200  more 

Myd. 

long. 

229. 
229. 
229. 
229 

1 
2 
3 

5 

$857.142f  A's.  $142.857|  7?'s. 
$480  As.  $750  B's.  $675  C's. 
$1500  Mr.  Ws.  $2100  Mr.  J's. 
$400  ^'s.  $800  B's.  $1200  C's. 

4 

($1866.66|  As. 
-]  $1066.66|  B's. 
($1066.66|  C's. 

332 


ANSWEKS. 


P. 

m 

230. 
230. 
230. 
230. 
230, 
230. 
230, 
230 


ANS. 


$77  A's.  $260  B's. 
$54.  As.  $38.50  B's. 

j$60.777+^'.9.    $127.633+J5's.    2 

|  $328.201 +  Z>'.s. 

$1666.66|  As.  $3888.88f  B's.  $9444.44f  C's. 
Rs.  =  $273.365  rcearfy.  ^'s= $476.635. 

(Fuller's  $1808.8669+,  Brown's  $1596.0591  + 

1  Dexter' s  $1995.0738+,  The  remainders  added 

( will  give  the  exact  proof. 


232. 

1 

$16.25 

7 

$8.93 

is* 

^2109.0392 

19 

$42.60 

232. 

2 

19.50.yd. 

8 

18.  06  step 

141 

575 

20 

4326ar. 

232. 

3 

39.375cto£. 

9 

$18.5487 

15! 

5229.08 

21 

42/zM. 

232. 

4 

$2.375 

10 

280  cows 

161 

;350 

22 

$24.25 

232 

5 

I55.48mi. 

11 

892.5  tons 

171 

^375 

232. 

6 

5  oxen. 

12 

1015/6. 

18j 

£694.232 

233. 

23 

$10.80 

1 

.25 

5 

.88A 

9 

•16* 

233 

91 

f  26f  per  ct.  left 

2 

.50 

6 

.05 

- 

233. 

\  —  3333.33$. 

3 

.40 

7 

:01A 

*>•-« 

25 

$1304  75 

4  j   20 

Q 

0^ 

235.  ||  2 

|  $24862.50  ||  3 

$233.75  | 

4  |  $8443.75)15  |  $14700 

236.  II  9  |  200  shares.  \\ 

237. 

1C 

) 

80  shares. 

238. 
238. 

l 

2 

I1.06J 

$0.75  loss. 

3 
4 

$0.966  + 
$1.00 

5 
6 

$112.50 

$208.4375 

7 
8 

12.054 
25  per  ct. 

239. 
239. 
239. 

9 
10 

18  per  c 
($13  w 

}        90 

t. 

hole  g'n 
oer  ct. 

11 
12 
13 

$1.025 
I1.03H. 

$2.216|. 

14 
15 

ISyd. 

$9.21TV 

(  —  ^U  i 

240. 
240. 
240. 
oin 

1 

2 

3 

$43.77 
$1312.50 
j  $237.60 
)  *1ft8  4.0 

4 
5 

6 

$210 
$607.50 
$1381.80  1 

*.^04. 

8  $450 
9  $1320 
0  $142.95 

11 
12 
13 
U 

$1800—  $45 
$47.624  + 
$9558,437  + 
*fiftno 

242. 

2 

$39 

.;; 

1427.50 

10 

$183,9705 

2!$121.325 

242. 

3 

$266 

7 

$9.5067 

11 

$4454.857 

3 

1315.389 

242. 

4 

$4446.75 

8 

$331.1511 

12  $30455.0224 

4 

221.075 

242 

5 

$642.60 

9 

$1158.0668 

l'$95.229  + 

5 

1290.798 

243.JI  2  |  $10.8012  j|  3  |  $2.728+         — 


ANSWERS. 


333 


p.   ||EX. 

ANS.              HEX.  |         ANS.            ||  EX 

ANS. 

244 
244. 

2 
3 

$309.5034 

$35.1485  + 

4 
5 

$30.5598 
$14.0979 

6 

$64.5792 

245. 
245. 
245. 
245. 
245. 
245. 

7 
8 
9 
10 
11 
12 

$76.2433 
$194.6177 
$328.32 
$1004.6976 
$1183.6935 
$1445.2333 

13 
14 
15 
16 
17 
181 

$190.148 
£3286.40 

£6322.8825 
£7500.60 
£75.04 
£218.88 

19 
20 
21 
22 
23 
24 

$600.445 
$44.2893 
$167.001 
$3126.203 
$9051.668 
$4968.9975 

246. 
246. 
246. 
246. 
246. 
246. 
247." 
247. 

25 
26 

27 
28 
29 
30 

$141.8136 
$272.80 
$39.9274 
$928.0686 
$529.925 
$31.2681 

31 
32 
33 
34 
35 
36 

$94.269 

$245.4896 
$76.966 
$33.3232 

$28761.776 
$5678.071 

37 
38 
39 
40 
41 
42 

$217.5116 
$6214.14 

$856.690 
$383.3808 
$188.0349 
$3720.465 

2 
3 

£15  2s.  8Jrf.              4 
£24  18s.  3Jd.+        5 

£26  10s.  11 
£331  Is.  Qa 

d 

i 

249. 

2 

$860.4194  ||  3  |  $167.983  + 

250.||1|$950||2|7  per  ^. 


251. 
251. 

2 
3 

$19.101 
$36.50  + 

4 
5 

$404.0625 
$291.60 

6 

7 

$211.456 

$185.775 

252 
253 
254 
254 
254 
254 
254 
255 
255 
256 
257 
258 


||8|$171.6Q75||9i$118.528||lQ|$315.2438||lli$152.408 
|  $1750 present  value.\\'2  \  $1565.402+  pres.  vol. 


254. 
254. 
254. 
254. 
254. 

3 

4 
5 
6 

7 

$9677.50+  pres.  val. 
£223  5s.  8d.  discount. 
$5620.176  +pres.  val. 
$702.485 
$1.94  difference. 

8 
9 
10 
11 
12 

$3869.407+  pres.  vol. 
$2109.236+     "       " 
$2763.694+     "       " 
$4000                " 
$6.473+  loss. 

255. 
255. 

1 
2 

$6.3291 
$10.50 

3 
4 

$15240.54 

$5.8408 

5 
6 

$3393.504 
$29.0096 

7 

$122.81  + 

256. 

8 

|  $341.709  +  ||1 

$344.66  +  ||2 

$5734.32  + 

|  $695.64||4|$118.85  +  |[5|$1740.60||6|376.46  + 
|2|12mo.l!3|87riQ. 


day  of  March. 


or 


ANSWERS. 


p. 


|EX.| 


ANS. 


EX 


ANS. 


2bo. 

1 

$426.416 

(£21  5s.-£25  14s.  3d  £30 

265. 

2 

£1073  18s.  l\d. 

6 

\  17s.  IcM 

1  2s.  9id  + 

265. 

3 

$1967.892  + 

(£38  11s. 

4K-£2319s.lUrf. 

265. 

4 

£389  6s.  2fd. 

K 

j  $250-$250-$250-$250. 

265. 

5 

$2551.733 

\  $516.66^ 

4250. 

266. 

1 

$3720.937 

3    $6748.60 

5 

$3643.875 

266. 

2 

$8668.935 

4    $4583.94  + 

- 



268. 
270T 


2  |  $1270.428  ||  3  ||  $2016.11    ||  4  |  $16975.775 


|2812.50||2 


3  |  1351.45+  ||  4 


271.111  I  3s.||2  | 


|  .288+c/s.||4 


|  73' 


274.! 
274J 
274. 
274. 
274 
274. 

3 

5 
6 
1 

2 
3 

1 
3 

3 
4 
1 
9 

4,  8,  2.   ||  4  |  1/6.  1/6.  3/6. 
of  16.  2  o/  18.  3  of  23.  5  o/  24 
jal  at  10s.-3  a/  14s.-4  at  21s.  4  a/  24s. 
gal  at  4s.,  4#a/.  at  5s.,  8  a/  5s.  Qd.,  ai 
46u.  TF.  286it.  E.  146it.  I?.  286ii.  0. 
66w.  W  1  2ft?/.  72.  1  2ft?/.  /?.  1  2ft?/,    O 

id  8  at  6s. 

275 

275' 
275 
275. 

4 
1 
1 

2 
3 

40(/a/.  F.  80^a/.  E.  20gal.  spirits. 
10  of  Is/.  10  of  2d.  30  of  M. 
36/6.  at  ±d.  36  at  6d.  36  at  Wd.  36  at  I2d. 
21  1  of  each. 
4  eac/i  of  the  1st.  three  and  30  of  15  cora/s  j£«e. 

276. 
276. 
276. 
276. 
276. 
276. 
276. 
276. 
276. 
276. 

1 

2 

3 
4 
5 
6 

7 
8 

I  =1 

J  -5-:hr. 
*=& 

9  =81 

123=1728 
1253  =  1953125 
163  =  4096 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 

94=6561 
165=1048576 
206  =  64000000 
2252=50625 
21672=4695889 
3213=33076161 
2154=2136750625 
=  610437195439776 
96=531441 
36()492  =  1299530401 

282. 

282. 
282 

28'I 

1 

2 
3 
4 
5 

1.732054- 
3,31662  + 
32.695  + 
1506.23  + 
2756.22  + 

6 
7 
8 
9 
.10 

6031 
4698 
57.19  + 
69.247  + 
2091  + 

11 

12 
IS 
H 

l« 

|.05 
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2.104 
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18 
19 
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0.71554 
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335 


284. 
284. 

2 
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25/35. 

1-26-4  9rd-  + 

3 
4 

85 
97.75mi.+ 

5 
6 

82  partners. 

Jp. 

28- 

28- 

285 

288 

288 

289 

289 

289 

290. 

292. 


EX. 


ANS. 


ANS. 


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ANS. 


7  |  62  trees  \\  8 


9  | 


10  |  4.90. 


288. 
288. 

1 

2 

73 

179 

3 
4 

319 
439 

5 
6 

638 
364 

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2 

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4    .909 

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1.505 

289. 

289. 
289. 

1 

2 
3 

i 

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t- 

4 
5 
6 

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.873  + 

1 
2 

3 

17 
28-4704 
16.197/35.+ 

4 

0 

6 

14.58/55.+ 
1728 
12/3L 

: 

6ft. 

290.  | 

8 

268.0832  ||  9 

2/15.  4iw.  ||  10  |  2ft. 

II  U 

12/2. 

j.  I  $1.53  ||  2  |  $212  ||  3  |  40  ||  4 
a     5mi.  II  f 


2      5 


^  )4")  !  i  ^ 

£2  2s.  8d. 

II  3  i  4  ||  4  I  78732  ||  5  |  $25600  ||  6  '  $61.44 

297. 

1 

6560 

3 

381 

5 

$196.83-$295.24 

297. 

2 

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120  men. 

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299. 

19 

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|  C"s  $32. 

31      -{    !  r,      /./ 

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2(J9. 

23 

7-^-  days. 

334 

299. 

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($2454  1st. 

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$4294.50  3d. 

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11  mo. 

300. 

39 

300  me?i 

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$3.653 

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5  years. 

336 


ANSWERS. 


p. 


H 


ANS. 


301. 

301. 
301. 
301. 
501. 
301. 
301. 
301. 
301. 
301. 


2250  men. 
(  $196.83  last  terms. 
j  $295.24  whole  ain't 
(  96fru.  wheat.  12  rye. 
1  12  barley.  12  oats. 


356.25 

$8640 

$1.20 

lost  4  pence. 


EX  | 

62 
63 
64 


ANS. 


4  days 
240  hour*. 
A  21-Ji  SC 


days. 


GY's—  $ 

Z>'*=$77.92}f 

$1020,66 

$8925.544  + 


302 
302 
302 
302 
302 
302 


50/£. 

9mi.  5/ur.  34rd.  9/£ 
j  daughter  $780.  so?i 
($3120.  wi/e$1560. 

76mi.-1292mi. 

4?/r.  llmo. 


$423.36 

$920.20  1st.  $2760.60 


74 

75j  |  2d.  5521.20  3d. 
76  3Ar.  20m. 
77j69fm.  f'n 


303. 
303. 
304, 
304 
304 
304 


400s#.  yd. 


305. 
305. 
305. 
305. 
305. 
305. 


5  10A 

6  7A  2 


QA.  OR.  12P. 


3  |  2 A  3P.  15P. 
1  !  109A  IJR.  28P. 


1 
2 
3 
4 
5 
6 


12.5664 

292.1688 

62.8320 

25. 

3709 

2180.41  + 


32520^.  yd. 
45849.485 
2  5A1P.9.95P. 
.  5'7"6"' 

28^2744 
78.5400 
38.4846 
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176.031258(?.  yd. 
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1809.5616 
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33.5104 
1436.7584 


2120.58 
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1242   6  3600   2  2290.2264  5  706.86 


309.1!  1 


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YB   17361 


M306011 


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THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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